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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>set </b><font color="Maroon">G</font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">pio</font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">poz</font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">US</font> =  <a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">LS</font> =  <a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <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="E2:46"/><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>and </b><br/><a NAME="E4:46"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>and </b><br/><a NAME="E5:46"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 ;<br/>

<b>set </b><font color="Maroon">UA</font> =  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">Wmin</font> =  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">Emax</font> =  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">Wbo</font> =  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">Ebo</font> =  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">Gij</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="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">Gik</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="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font>;<br/>
<b>set </b><font color="Maroon">Gi1k</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="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font>;<br/>
<a NAME="E6:46"/><i><font color="Green">E47</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <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> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <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>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:46"/><i><font color="Green">E48</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal3.html#T23">TOPREAL3:23</a></i>;<br/>
<a NAME="E8:46"/>
<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>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:46"/><b>then </b><i><font color="Green">E50</font></i>: 
<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>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:46"/>
<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>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:46"/><b>then </b><i><font color="Green">E51</font></i>: 
<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>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:46"/><i><font color="Green">E73</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:46"/><i><font color="Green">E74</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:46"/><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan15.html#E3:46"><i><font color="Green">E42</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E15:46"/><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E16:46"/><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E17:46"/><i><font color="Green">E78</font></i>: 
 <a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</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>
 
<b>by </b><i><a class="ref" href="jordan1g.html#T4">JORDAN1G:4</a></i>;<br/>
<a NAME="E18:46"/><i><font color="Green">E79</font></i>: 
 <a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</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>
 
<b>by </b><i><a class="ref" href="jordan1g.html#T5">JORDAN1G:5</a></i>;<br/>
<b>set </b><font color="Maroon">go</font> =  <a href="jordan3.html#K4">R_Cut</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">do</font> =  <a href="jordan3.html#K3">L_Cut</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>;<br/>
<a NAME="E19:46"/><i><font color="Green">E83</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</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>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E20:46"/><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 
<b>by </b><i><a class="ref" href="jordan1e.html#T19">JORDAN1E:19</a></i>;<br/>
<a NAME="E21:46"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E22:46"/><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"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E85</font></i>: <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1f.html#T5">JORDAN1F:5</a></i>
;<br/>
<i><font color="Green">E86</font></i>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  = 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="jordan1a.html#T94">JORDAN1A:94</a></i>
;<br/>
<a NAME="E25:46"/>
 <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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 
<b>by </b><i><a class="ref" href="jordan8.html#T13">JORDAN8:13</a></i>;<br/>
<a NAME="E26:46"/><b>then </b><i><font color="Green">E87</font></i>: 
 <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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E27:46"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E28:46"/><b>then </b><i><font color="Green">E90</font></i>: 
<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>,<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i>, <a class="txt" href="jordan15.html#E5:46"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan15.html#E11:46"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan15.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">go</font> =  <a href="jordan3.html#K4">R_Cut</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> as  <a href="topreal1.html#V4">being_S-Seq</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan15.html#E2:46"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jordan3.html#T70">JORDAN3:70</a></i>;<br/>
<a NAME="E30:46"/><i><font color="Green">E91</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="jordan1e.html#T19">JORDAN1E:19</a></i>;<br/>
<a NAME="E31:46"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E32:46"/><b>then </b><i><font color="Green">E92</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E93</font></i>: <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1f.html#T8">JORDAN1F:8</a></i>
;<br/>
<i><font color="Green">E96</font></i>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  = 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="jordan1a.html#T94">JORDAN1A:94</a></i>
;<br/>
<a NAME="E35:46"/><i><font color="Green">E97</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan15.html#E3:46"><i><font color="Green">E42</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E36:46"/><b>then </b><i><font color="Green">E98</font></i>: 
<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>,<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan15.html#T4" target="_self">Th4</a>, , , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">do</font> =  <a href="jordan3.html#K3">L_Cut</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> as  <a href="topreal1.html#V4">being_S-Seq</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan15.html#E1:46"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan3.html#T69">JORDAN3:69</a></i>;<br/>
<a NAME="E38:46"/><i><font color="Green">E99</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</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>,<font color="Maroon">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<i><font color="Green">E100</font></i>: <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1f.html#T6">JORDAN1F:6</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  = 
 <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</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>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="jordan1a.html#T92">JORDAN1A:92</a></i>
;<br/>
<a NAME="E41:46"/><b>then </b><i><font color="Green">E102</font></i>: 
<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>,<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan15.html#T6" target="_self">Th6</a>, , , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/>
<a NAME="E42:46"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E43:46"/><i><font color="Green">E105</font></i>: 
<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>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan15.html#E2:46"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan15.html#E6:46"><i><font color="Green">E47</font></i></a>, <a class="ref" href="jordan1j.html#T40">JORDAN1J:40</a></i>;<br/>
<a NAME="E44:46"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">go</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>
 
<b>by </b><i><a class="txt" href="jordan15.html#E6:46"><i><font color="Green">E47</font></i></a>, <a class="ref" href="jordan1j.html#T38">JORDAN1J:38</a></i>;<br/>
<a NAME="E45:46"/><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">do</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E46:46"/><i><font color="Green">E108</font></i>: 
<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>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan15.html#E1:46"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan15.html#E3:46"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan15.html#E7:46"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan1j.html#T40">JORDAN1J:40</a></i>;<br/>
<a NAME="E47:46"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">do</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>
 
<b>by </b><i><a class="txt" href="jordan15.html#E7:46"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan1j.html#T39">JORDAN1J:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon">go</font> = <font color="Maroon">go</font> as  non <a href="funct_1.html#V3">constant</a> <a href="topreal1.html#V4">being_S-Seq</a> <a href="goboard5.html#V1">s.c.c.</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, , <a class="ref" href="jgraph_1.html#T16">JGRAPH_1:16</a>, <a class="ref" href="jordan8.html#T8">JORDAN8:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">do</font> = <font color="Maroon">do</font> as  non <a href="funct_1.html#V3">constant</a> <a href="topreal1.html#V4">being_S-Seq</a> <a href="goboard5.html#V1">s.c.c.</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan15.html#T8" target="_self">Th8</a>, , <a class="ref" href="jgraph_1.html#T16">JGRAPH_1:16</a>, <a class="ref" href="jordan8.html#T8">JORDAN8:8</a></i>;<br/>
<a NAME="E50:46"/><i><font color="Green">E110</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E51:46"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">go</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E112</font></i>: <font color="Maroon">go</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">go</font></span>)</span> = 
<font color="Maroon">go</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<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>,<font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="jordan15.html#E2:46"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jordan3.html#T59">JORDAN3:59</a></i>
;<br/>
<a NAME="E53:46"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">do</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="jordan15.html#T8" target="_self">Th8</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E54:46"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">do</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E113</font></i>: <font color="Maroon">do</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">do</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<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>,<font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="jordan15.html#E1:46"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan3.html#T58">JORDAN3:58</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_3.html#T28">FINSEQ_3:28</a></i>;<br/>
<a NAME="E57:46"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font>
 
;<br/>
<a NAME="E58:46"/><b>then </b><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E59:46"/><i><font color="Green">E116</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<font color="Maroon">m</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font>
 
<b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/>
<a NAME="E60:46"/><i><font color="Green">E117</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E2:46"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jordan3.html#T76">JORDAN3:76</a></i>;<br/>
<a NAME="E61:46"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<font color="Maroon">m</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E16:46"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E62:46"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<font color="Maroon">m</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E63:46"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E64:46"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E12:46"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan15.html#E14:46"><i><font color="Green">E75</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E65:46"/>
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<font color="Maroon">m</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_9"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/>

<a NAME="E2:46_9"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">x</font> <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>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:46_9"/><i><font color="Green">E121</font></i>: 
<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>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E4:46_9"/>
<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>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:46_9"/><b>then </b>
<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>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:46_9"/><b>then </b>
<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>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T10">SPRECT_1:10</a></i>;<br/>
<b>hence </b><a NAME="E7:46_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<font color="Maroon">m</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E66:46"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<font color="Maroon">m</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E18:46"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E67:46"/><i><font color="Green">E123</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1 <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font>
 
<b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/>
<a NAME="E68:46"/><i><font color="Green">E124</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E1:46"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan3.html#T77">JORDAN3:77</a></i>;<br/>
<a NAME="E69:46"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1 <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E70:46"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E71:46"/><i><font color="Green">E126</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">do</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan15.html#T8" target="_self">Th8</a>, <a class="txt" href="jordan15.html#E13:46"><i><font color="Green">E74</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E72:46"/>
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="realset1.html#K5">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_10"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_10"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/>

<a NAME="E2:46_10"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">x</font> <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>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:46_10"/><i><font color="Green">E128</font></i>: 
<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>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1
 
<b>by </b><i><a class="txt" href="jordan15.html#E23:46"><i><font color="Green">E85</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E4:46_10"/>
<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>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:46_10"/><b>then </b>
<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>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:46_10"/><b>then </b>
<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>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T10">SPRECT_1:10</a></i>;<br/>
<b>hence </b><a NAME="E7:46_10"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E24:46"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan15.html#E26:46"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E73:46"/><b>then </b><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E20:46"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<i><font color="Green">E130</font></i>: <font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan15.html#E2:46"><i><font color="Green">E41</font></i></a>, <a class="ref" href="sprect_3.html#T39">SPRECT_3:39</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1f.html#T5">JORDAN1F:5</a></i>
;<br/>
<b>then </b><i><font color="Green">E131</font></i>: <font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1f.html#T8">JORDAN1F:8</a></i>
<br/>.= 
<font color="Maroon">do</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">do</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan15.html#E1:46"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan1j.html#T35">JORDAN1J:35</a></i>
;<br/>
<a NAME="E76:46"/><i><font color="Green">E134</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">go</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">do</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font> )
 
<b>by </b><i>, <a class="ref" href="jordan15.html#T8" target="_self">Th8</a>, <a class="ref" href="sppol_2.html#T18">SPPOL_2:18</a></i>;<br/>
<a NAME="E77:46"/><i><font color="Green">E156</font></i>: 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_13"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_13"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/>

<a NAME="E2:46_13"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:46_13"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">go</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">do</font> )
 
<b>by </b><i><a class="txt" href="jordan15.html#E27:46"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_6.html#T46">FINSEQ_6:46</a>, <a class="ref" href="revrot_1.html#T3">REVROT_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:46_13"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E157</font></i>: <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1f.html#T6">JORDAN1F:6</a></i>
;<br/>
<a NAME="E79:46"/><i><font color="Green">E158</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</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>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E3:46"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E80:46"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_15"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_15"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span>
 ;<br/>

<a NAME="E2:46_15"/><b>then </b><i><font color="Green">E159</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:46_15"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E17:46"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan15.html#E19:46"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:46_15"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="jordan1e.html#T20">JORDAN1E:20</a></i>;<br/>
<a NAME="E5:46_15"/><b>then </b><i><font color="Green">E160</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_15_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:46_15_1"/><b>then </b><i><font color="Green">E161</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <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>,<font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E1:46"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan15.html#E30:46"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan15.html#E33:46"><i><font color="Green">E93</font></i></a>, <a class="ref" href="jordan1e.html#T11">JORDAN1E:11</a></i>;<br/><a NAME="E3:46_15_1"/>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</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>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E3:46"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="jordan1a.html#T92">JORDAN1A:92</a></i>;<br/><a NAME="E4:46_15_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan15.html#E32:46"><i><font color="Green">E92</font></i></a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><b>hence </b><a NAME="E5:46_15_1"/>
contradiction
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:46_15"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="txt" href="jordan15.html#E26:46"><i><font color="Green">E87</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E81:46"/><b>then </b><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E28:46"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">W2</font> = <font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> 2;<br/>
<a NAME="E82:46"/><i><font color="Green">E164</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">go</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E83:46"/><i><font color="Green">E165</font></i>: 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/>
<div><i><font color="Green">E166</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_16"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_16"/>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:46_16"/><b>then </b>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="jordan1a.html#T94">JORDAN1A:94</a></i>;<br/><b>hence </b><a NAME="E3:46_16"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan15.html#E5:46"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan15.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div>
<font color="Maroon">go</font> = 
 <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan1g.html#T57">JORDAN1G:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>
;<br/>
<a NAME="E86:46"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>
<a NAME="E87:46"/><i><font color="Green">E169</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">go</font>
 
<b>by </b><i><a class="txt" href="jordan15.html#E26:46"><i><font color="Green">E87</font></i></a>, <a class="ref" href="finseq_6.html#T46">FINSEQ_6:46</a></i>;<br/>
<b>set </b><font color="Maroon">pion</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>;<br/>
<div><i><font color="Green">E170</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_18"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:46_18"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 ;<br/><a NAME="E2:46_18"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a>, <a class="ref" href="finseq_3.html#T30">FINSEQ_3:30</a></i>;<br/><a NAME="E3:46_18"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><a NAME="E4:46_18"/><b>then </b>
( <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <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>,<font color="Maroon">k</font> or <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <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>,<font color="Maroon">k</font> or <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <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>,<font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>;<br/><b>hence </b><a NAME="E5:46_18"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <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="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i>, <a class="txt" href="jordan15.html#E5:46"><i><font color="Green">E45</font></i></a>, </i>;<br/></div><b>end;</b></div>
<i><font color="Green">E172</font></i>: <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, <a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, <a class="txt" href="jordan15.html#E3:46"><i><font color="Green">E42</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/>
<a NAME="E91:46"/><b>then </b><i><font color="Green">E173</font></i>: 
<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>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E39:46"><i><font color="Green">E100</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E92:46"/><i><font color="Green">E174</font></i>: 
<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>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E93:46"/><i><font color="Green">E175</font></i>: 
<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>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_21_1"/>
( ( <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a>  ) or <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1"><b>suppose </b></a><a NAME="E1:46_21_1_1"/>
( <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:46_21_1_1"/><b>then </b>
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 <b>by </b><i>, <a class="ref" href="topreal3.html#T42">TOPREAL3:42</a></i>;<br/><b>then consider </b><font color="Maroon">pion1</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:46_21_1_1"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">pion1</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>
 <b>and </b><br/><a NAME="E5:46_21_1_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">pion1</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 <b>and </b><br/><a NAME="E6:46_21_1_1"/><i><font color="Green">E179</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font>
 <b>and </b><br/><a NAME="E7:46_21_1_1"/><i><font color="Green">E180</font></i>: 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E8:46_21_1_1"/><i><font color="Green">E181</font></i>: 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">pion1</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">pion1</font></span>)</span>
 <b>and </b><br/><a NAME="E9:46_21_1_1"/><i><font color="Green">E182</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E38:46"><i><font color="Green">E99</font></i></a>, <a class="ref" href="goboard3.html#T2">GOBOARD3:2</a></i>;<br/><b>reconsider </b><font color="Maroon">pion1</font> = <font color="Maroon">pion1</font> as  <a href="topreal1.html#V4">being_S-Seq</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan15.html#E43:46"><i><font color="Green">E105</font></i></a></i>;<br/><b>set </b><font color="Maroon">godo</font> = <span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="graph_2.html#K4">^'</a> <font color="Maroon">do</font>;<br/><i><font color="Green">E183</font></i>: <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, <a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, <a class="txt" href="jordan15.html#E3:46"><i><font color="Green">E42</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/><a NAME="E12:46_21_1_1"/><i><font color="Green">E184</font></i>: 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan1a.html#T39">JORDAN1A:39</a></i>;<br/><a NAME="E13:46_21_1_1"/><b>then </b><i><font color="Green">E185</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="sprect_1.html#T62">SPRECT_1:62</a></i>;<br/><a NAME="E14:46_21_1_1"/><i><font color="Green">E186</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan15.html#E50:46"><i><font color="Green">E110</font></i></a>, <a class="ref" href="sprect_1.html#T62">SPRECT_1:62</a></i>;<br/> <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sprect_1.html#T54">SPRECT_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E50:46"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan15.html#E51:46"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan15.html#E52:46"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan15.html#E55:46"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/><a NAME="E16:46_21_1_1"/><b>then </b><i><font color="Green">E187</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan15.html#E44:46"><i><font color="Green">E106</font></i></a>, <a class="ref" href="topreal3.html#T23">TOPREAL3:23</a></i>;<br/><a NAME="E17:46_21_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <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/><a NAME="E18:46_21_1_1"/><b>then </b><i><font color="Green">E188</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:46_21_1_1"/><b>then </b><i><font color="Green">E189</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="revrot_1.html#T14">REVROT_1:14</a></i>;<br/><a NAME="E20:46_21_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font>
 <b>by </b><i><a class="ref" href="topreal8.html#T7">TOPREAL8:7</a></i>;<br/><a NAME="E21:46_21_1_1"/><b>then </b><i><font color="Green">E190</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E22:46_21_1_1"/><b>then </b><i><font color="Green">E191</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E23:46_21_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="graph_2.html#K4">^'</a> <font color="Maroon">do</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal8.html#T7">TOPREAL8:7</a></i>;<br/><a NAME="E24:46_21_1_1"/><b>then </b><i><font color="Green">E192</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="graph_2.html#K4">^'</a> <font color="Maroon">do</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E60:46"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E25:46_21_1_1"/><i><font color="Green">E193</font></i>: 
 <a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</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>
 <b>by </b><i><a class="ref" href="jordan1g.html#T4">JORDAN1G:4</a></i>;<br/><a NAME="E26:46_21_1_1"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">go</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">go</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E12:46"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan15.html#E45:46"><i><font color="Green">E107</font></i></a>, <a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>;<br/><a NAME="E27:46_21_1_1"/><b>then </b><i><font color="Green">E195</font></i>: 
<font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</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>
 <b>by </b><i>, <a class="ref" href="jordan15.html#T13" target="_self">Th13</a>, <a class="ref" href="topreal8.html#T12">TOPREAL8:12</a></i>;<br/><i><font color="Green">E196</font></i>: <span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span></span>)</span> = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="jordan15.html#E46:46"><i><font color="Green">E108</font></i></a>, <a class="ref" href="graph_2.html#T58">GRAPH_2:58</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
<font color="Maroon">do</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan15.html#E13:46"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/><a NAME="E29:46_21_1_1"/><b>then </b><i><font color="Green">E197</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="graph_2.html#K4">^'</a> <font color="Maroon">do</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>
 <b>by </b><i>, <a class="txt" href="jordan15.html#E66:46"><i><font color="Green">E122</font></i></a>, <a class="ref" href="topreal8.html#T12">TOPREAL8:12</a></i>;<br/><a NAME="E30:46_21_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,1 <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan15.html#E44:46"><i><font color="Green">E106</font></i></a>, <a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/><a NAME="E31:46_21_1_1"/><b>then </b><i><font color="Green">E198</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="txt" href="jordan15.html#E15:46"><i><font color="Green">E76</font></i></a>, , <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E32:46_21_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E47:46"><i><font color="Green">E109</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E33:46_21_1_1"/><b>then </b><i><font color="Green">E199</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E34:46_21_1_1"/>
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<font color="Maroon">m</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_21_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/>

<a NAME="E2:46_21_1_1_4"/><b>then </b><i><font color="Green">E200</font></i>: 
<font color="Maroon">x</font> <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>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:46_21_1_1_4"/><i><font color="Green">E201</font></i>: 
<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>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E4:46_21_1_1_4"/>
<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>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,1
 
<b>by </b><i><a class="txt" href="jordan15.html#E12:46"><i><font color="Green">E73</font></i></a>, , <a class="txt" href="jordan15.html#E71:46"><i><font color="Green">E126</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/>
<b>hence </b><a NAME="E5:46_21_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<font color="Maroon">m</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:46_21_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">go</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font></span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="txt" href="jordan15.html#E12:46"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan15.html#E15:46"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan15.html#E70:46"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E36:46_21_1_1"/><b>then </b><i><font color="Green">E202</font></i>: 
<font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, <a class="ref" href="topreal8.html#T34">TOPREAL8:34</a></i>;<br/><a NAME="E37:46_21_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E47:46"><i><font color="Green">E109</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E38:46_21_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E39:46_21_1_1"/><b>then </b><i><font color="Green">E203</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E40:46_21_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#T13">GRAPH_2:13</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E42:46_21_1_1"/><b>then </b><i><font color="Green">E204</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E43:46_21_1_1"/><i><font color="Green">E205</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E71:46"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E44:46_21_1_1"/><b>then </b><i><font color="Green">E206</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E45:46_21_1_1"/><b>then </b><i><font color="Green">E207</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><i><font color="Green">E208</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan15.html#E75:46"><i><font color="Green">E131</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E47:46_21_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font>
 ;<br/><a NAME="E48:46_21_1_1"/><b>then </b><i><font color="Green">E209</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font>
 <b>by </b><i><a class="ref" href="jordan15.html#T15" target="_self">Th15</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E49:46_21_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan15.html#E44:46"><i><font color="Green">E106</font></i></a>, <a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/><a NAME="E50:46_21_1_1"/><b>then </b><i><font color="Green">E210</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="txt" href="jordan15.html#E24:46"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E51:46_21_1_1"/>
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="realset1.html#K5">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_21_1_1_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/>

<a NAME="E2:46_21_1_1_7"/><b>then </b><i><font color="Green">E211</font></i>: 
<font color="Maroon">x</font> <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>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:46_21_1_1_7"/><i><font color="Green">E212</font></i>: 
<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>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1
 
<b>by </b><i><a class="txt" href="jordan15.html#E23:46"><i><font color="Green">E85</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 3
<b>by </b><i><a class="txt" href="jordan15.html#E46:46"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan15.html#T15" target="_self">Th15</a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
<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>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/>
<a NAME="E5:46_21_1_1_7"/><b>then </b>
<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>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E74:46"><i><font color="Green">E130</font></i></a>, <a class="ref" href="jordan15.html#T15" target="_self">Th15</a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/>
<b>hence </b><a NAME="E6:46_21_1_1_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan15.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan15.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E52:46_21_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">pion1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="ref" href="jordan15.html#T20" target="_self">Th20</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E53:46_21_1_1"/><b>then </b><i><font color="Green">E213</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">go</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span></span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="txt" href="jordan15.html#E13:46"><i><font color="Green">E74</font></i></a>, , <a class="txt" href="jordan15.html#E67:46"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jordan15.html#T16" target="_self">Th16</a>, <a class="txt" href="jordan15.html#E76:46"><i><font color="Green">E134</font></i></a>, <a class="ref" href="topreal8.html#T31">TOPREAL8:31</a></i>;<br/><a NAME="E54:46_21_1_1"/><i><font color="Green">E214</font></i>: 
not <font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="jordan15.html#E60:46"><i><font color="Green">E117</font></i></a>, <a class="ref" href="realset1.html#T13">REALSET1:13</a></i>;<br/><a NAME="E55:46_21_1_1"/><i><font color="Green">E215</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">pion1</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E71:46"><i><font color="Green">E126</font></i></a>, <a class="ref" href="sppol_2.html#T18">SPPOL_2:18</a></i>;<br/><a NAME="E56:46_21_1_1"/><i><font color="Green">E216</font></i>: 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_8"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_21_1_1_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/>

<a NAME="E2:46_21_1_1_8"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:46_21_1_1_8"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">go</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">pion1</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_6.html#T46">FINSEQ_6:46</a>, <a class="ref" href="revrot_1.html#T3">REVROT_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:46_21_1_1_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan15.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E57:46_21_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_9"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_21_1_1_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span>
 ;<br/>

<a NAME="E2:46_21_1_1_9"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:46_21_1_1_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i>, , <a class="txt" href="jordan15.html#E12:46"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan15.html#E17:46"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan15.html#E44:46"><i><font color="Green">E106</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E58:46_21_1_1"/><b>then </b><i><font color="Green">E217</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="ref" href="jordan15.html#T25" target="_self">Th25</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E59:46_21_1_1"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font> is <a href="topreal1.html#V3">s.n.c.</a>
 <b>by </b><i>, <a class="ref" href="jordan1j.html#T54">JORDAN1J:54</a></i>;<br/><a NAME="E60:46_21_1_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">pion1</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i>, <a class="ref" href="jordan15.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan15.html#T26" target="_self">Th26</a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E61:46_21_1_1"/><b>then </b><i><font color="Green">E220</font></i>: 
<font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="jordan1j.html#T55">JORDAN1J:55</a></i>;<br/><i><font color="Green">E221</font></i>: <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
<font color="Maroon">do</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan15.html#E13:46"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/><a NAME="E63:46_21_1_1"/><i><font color="Green">E222</font></i>: 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_11"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_21_1_1_11"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/>

<a NAME="E2:46_21_1_1_11"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pion1</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">pion1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:46_21_1_1_11"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">do</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">pion1</font> )
 
<b>by </b><i><a class="txt" href="jordan15.html#E46:46"><i><font color="Green">E108</font></i></a>, , <a class="ref" href="finseq_6.html#T46">FINSEQ_6:46</a>, <a class="ref" href="revrot_1.html#T3">REVROT_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:46_21_1_1_11"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan15.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E64:46_21_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_12"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_21_1_1_12"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span>
 ;<br/>

<a NAME="E2:46_21_1_1_12"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:46_21_1_1_12"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i>, , <a class="txt" href="jordan15.html#E13:46"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan15.html#E19:46"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan15.html#E44:46"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan15.html#E46:46"><i><font color="Green">E108</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E65:46_21_1_1"/><b>then </b><i><font color="Green">E223</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pion1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">pion1</font></span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><i><font color="Green">E224</font></i>: <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span>
<b>by </b><i>, <a class="ref" href="topreal8.html#T35">TOPREAL8:35</a></i>
<br/>.= 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">do</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
<b>by </b><i><a class="txt" href="jordan15.html#E33:46"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jordan15.html#E46:46"><i><font color="Green">E108</font></i></a>, , , <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">do</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
<b>by </b><i><a class="ref" href="graph_2.html#T57">GRAPH_2:57</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">do</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
;<br/><a NAME="E67:46_21_1_1"/>
<font color="Maroon">do</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">do</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E27:46"><i><font color="Green">E88</font></i></a>, <a class="ref" href="graph_2.html#T57">GRAPH_2:57</a></i>;<br/><b>then reconsider </b><font color="Maroon">godo</font> = <span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="graph_2.html#K4">^'</a> <font color="Maroon">do</font> as  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a> <b>by </b><i><a class="txt" href="jordan15.html#E64:46"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan15.html#E67:46"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan15.html#E68:46"><i><font color="Green">E124</font></i></a>, , <a class="txt" href="jordan15.html#E73:46"><i><font color="Green">E129</font></i></a>, <a class="ref" href="jordan15.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan15.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan15.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan15.html#T30" target="_self">Th30</a>, , <a class="ref" href="jordan8.html#T7">JORDAN8:7</a>, <a class="ref" href="jordan8.html#T8">JORDAN8:8</a>, <a class="ref" href="topreal8.html#T11">TOPREAL8:11</a>, <a class="ref" href="topreal8.html#T33">TOPREAL8:33</a>, <a class="ref" href="topreal8.html#T34">TOPREAL8:34</a></i>;<br/><a NAME="E69:46_21_1_1"/><i><font color="Green">E225</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan6.html#D8">JORDAN6:def 8</a></i>;<br/><a NAME="E70:46_21_1_1"/><b>then </b><i><font color="Green">E226</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T11">JORDAN6:11</a></i>;<br/><a NAME="E71:46_21_1_1"/><i><font color="Green">E227</font></i>: 
(  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> &amp;  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/><b>set </b><font color="Maroon">ff</font> =  <a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E72:46_21_1_1"/>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_2.html#T47">SPRECT_2:47</a></i>;<br/><a NAME="E73:46_21_1_1"/><b>then </b><i><font color="Green">E228</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T98">FINSEQ_6:98</a></i>;<br/><a NAME="E74:46_21_1_1"/><i><font color="Green">E229</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="revrot_1.html#T33">REVROT_1:33</a></i>;<br/><a NAME="E75:46_21_1_1"/><b>then </b><i><font color="Green">E230</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="sprect_5.html#T23">SPRECT_5:23</a></i>;<br/><a NAME="E76:46_21_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="sprect_5.html#T24">SPRECT_5:24</a></i>;<br/><a NAME="E77:46_21_1_1"/><b>then </b><i><font color="Green">E231</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E78:46_21_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="sprect_5.html#T25">SPRECT_5:25</a></i>;<br/><a NAME="E79:46_21_1_1"/><b>then </b><i><font color="Green">E232</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E80:46_21_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="sprect_5.html#T26">SPRECT_5:26</a></i>;<br/><a NAME="E81:46_21_1_1"/><b>then </b><i><font color="Green">E233</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><i><font color="Green">E234</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_14"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_21_1_1_14"/><i><font color="Green">E235</font></i>: 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><a NAME="E2:46_21_1_1_14"/>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/><a NAME="E3:46_21_1_1_14"/><b>then </b>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E78:46"><i><font color="Green">E157</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:46_21_1_1_14"/><b>then </b>
<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>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_5.html#T41">FINSEQ_5:41</a></i>;<br/><b>hence </b><a NAME="E5:46_21_1_1_14"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan15.html#E11:46"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan15.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan1f.html#T5">JORDAN1F:5</a></i>;<br/></div><b>end;</b></div><a NAME="E83:46_21_1_1"/><i><font color="Green">E236</font></i>: 
 <a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</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>
 <b>by </b><i><a class="ref" href="jordan9.html#D1">JORDAN9:def 1</a></i>;<br/><a NAME="E84:46_21_1_1"/><b>then </b><i><font color="Green">E237</font></i>: 
 <a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <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>
 <b>by </b><i><a class="ref" href="revrot_1.html#T34">REVROT_1:34</a></i>;<br/><a NAME="E85:46_21_1_1"/><i><font color="Green">E238</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">godo</font>,1,<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> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">godo</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E64:46"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan15.html#E68:46"><i><font color="Green">E124</font></i></a>, <a class="ref" href="jordan9.html#T29">JORDAN9:29</a></i>;<br/><i><font color="Green">E239</font></i>:  <a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan15.html#E67:46"><i><font color="Green">E123</font></i></a>, <a class="ref" href="topreal8.html#T35">TOPREAL8:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span>
<b>by </b><i>, <a class="ref" href="topreal8.html#T35">TOPREAL8:35</a></i>
;<br/><a NAME="E87:46_21_1_1"/>
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan1e.html#T17">JORDAN1E:17</a></i>;<br/><a NAME="E88:46_21_1_1"/><b>then </b><i><font color="Green">E240</font></i>: 
(  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E89:46_21_1_1"/><b>then </b><i><font color="Green">E241</font></i>: 
(  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan15.html#E17:46"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan15.html#E19:46"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E90:46_21_1_1"/><i><font color="Green">E242</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="sprect_1.html#T15">SPRECT_1:15</a></i>;<br/><div><i><font color="Green">E243</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_16"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_21_1_1_16"/>
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font>
 ;<br/><a NAME="E2:46_21_1_1_16"/><b>then </b><i><font color="Green">E244</font></i>: 
(  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span> or  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_21_1_1_16_1"/>
(  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font> or  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font> or  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font> )
 </span><b>by </b><i><a class="txt" href="jordan15.html#E83:46"><i><font color="Green">E165</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_16_1_1"><b>suppose </b></a><a NAME="E1:46_21_1_1_16_1_1"/>
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font>
 ;<br/><div class="add"><a NAME="E2:46_21_1_1_16_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan15.html#E81:46"><i><font color="Green">E163</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:46_21_1_1_16_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan10.html#T5">JORDAN10:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_16_1_2"><b>suppose </b></a><a NAME="E1:46_21_1_1_16_1_2"/>
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_21_1_1_16_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan15.html#E44:46"><i><font color="Green">E106</font></i></a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_16_1_3"><b>suppose </b></a><a NAME="E1:46_21_1_1_16_1_3"/>
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font>
 ;<br/><div class="add"><a NAME="E2:46_21_1_1_16_1_3"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan15.html#E81:46"><i><font color="Green">E163</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:46_21_1_1_16_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan10.html#T5">JORDAN10:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div> <a href="goboard5.html#K4">right_cell</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1 = 
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan15.html#E59:46"><i><font color="Green">E116</font></i></a>, <a class="ref" href="jordan1h.html#T29">JORDAN1H:29</a></i>
<br/>.= 
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="revrot_1.html#T28">REVROT_1:28</a></i>
<br/>.= 
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1,<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="jordan1h.html#T52">JORDAN1H:52</a></i>
<br/>.= 
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_5.html#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1,<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="txt" href="jordan15.html#E79:46"><i><font color="Green">E158</font></i></a>, <a class="ref" href="jordan1j.html#T53">JORDAN1J:53</a></i>
<br/>.= 
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,1,<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="jordan1e.html#D1">JORDAN1E:def 1</a></i>
<br/>.= 
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4">R_Cut</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,1,<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="txt" href="jordan15.html#E77:46"><i><font color="Green">E156</font></i></a>, <a class="ref" href="jordan1j.html#T52">JORDAN1J:52</a></i>
<br/>.= 
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span>,1,<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="txt" href="jordan15.html#E66:46"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jordan1j.html#T51">JORDAN1J:51</a></i>
<br/>.= 
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">godo</font>,1,<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="txt" href="jordan15.html#E62:46"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan15.html#E68:46"><i><font color="Green">E124</font></i></a>, <a class="ref" href="jordan1j.html#T51">JORDAN1J:51</a></i>
;<br/><a NAME="E93:46_21_1_1"/><b>then </b>
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">godo</font>,1,<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="jordan1i.html#T8">JORDAN1I:8</a></i>;<br/><a NAME="E94:46_21_1_1"/><b>then </b><i><font color="Green">E245</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">godo</font>,1,<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> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E82:46"><i><font color="Green">E164</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><i><font color="Green">E246</font></i>: <font color="Maroon">godo</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="graph_2.html#T57">GRAPH_2:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan15.html#E26:46"><i><font color="Green">E87</font></i></a>, <a class="ref" href="graph_2.html#T57">GRAPH_2:57</a></i>
;<br/><a NAME="E96:46_21_1_1"/><i><font color="Green">E247</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="txt" href="jordan15.html#E9:46"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><i><font color="Green">E248</font></i>: <font color="Maroon">godo</font> <a href="finseq_4.html#K4">/.</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon">go</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">pion1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan15.html#E60:46"><i><font color="Green">E117</font></i></a>, <a class="ref" href="graph_2.html#T61">GRAPH_2:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i>, <a class="txt" href="jordan15.html#E35:46"><i><font color="Green">E97</font></i></a>, <a class="ref" href="graph_2.html#T61">GRAPH_2:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="graph_2.html#K4">^'</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan15.html#E86:46"><i><font color="Green">E167</font></i></a>, <a class="ref" href="graph_2.html#T61">GRAPH_2:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="ref" href="jordan1e.html#T15">JORDAN1E:15</a></i>
;<br/><a NAME="E98:46_21_1_1"/><i><font color="Green">E249</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T19">COMPTS_1:19</a></i>;<br/><a NAME="E99:46_21_1_1"/><i><font color="Green">E250</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E100:46_21_1_1"/>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan15.html#E36:46"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E101:46_21_1_1"/><b>then </b><i><font color="Green">E251</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E87:46"><i><font color="Green">E169</font></i></a>, <a class="txt" href="jordan15.html#E88:46"><i><font color="Green">E170</font></i></a>, <a class="ref" href="jordan1j.html#T21">JORDAN1J:21</a></i>;<br/><a NAME="E102:46_21_1_1"/><i><font color="Green">E252</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E103:46_21_1_1"/>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E2:46"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/><a NAME="E104:46_21_1_1"/><b>then </b>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E34:46"><i><font color="Green">E96</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E105:46_21_1_1"/><b>then </b>
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E57:46"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan15.html#E87:46"><i><font color="Green">E169</font></i></a>, , <a class="txt" href="jordan15.html#E89:46"><i><font color="Green">E172</font></i></a>, <a class="ref" href="jordan1j.html#T33">JORDAN1J:33</a></i>;<br/><a NAME="E106:46_21_1_1"/><b>then </b><i><font color="Green">E253</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E107:46_21_1_1"/><i><font color="Green">E254</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">godo</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E64:46"><i><font color="Green">E119</font></i></a>, <a class="ref" href="sppol_2.html#T18">SPPOL_2:18</a></i>;<br/><a NAME="E108:46_21_1_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">godo</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E64:46"><i><font color="Green">E119</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E109:46_21_1_1"/><b>then </b><i><font color="Green">E255</font></i>: 
<font color="Maroon">godo</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">godo</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E110:46_21_1_1"/>
<font color="Maroon">godo</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K23">W-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan1i.html#T27">JORDAN1I:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">godo</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E91:46"><i><font color="Green">E173</font></i></a>, <a class="ref" href="pscomp_1.html#T88">PSCOMP_1:88</a></i>
<br/>.= 
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><a NAME="E112:46_21_1_1"/><b>then </b>
<font color="Maroon">godo</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K23">W-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E92:46"><i><font color="Green">E174</font></i></a>, <a class="txt" href="jordan15.html#E93:46"><i><font color="Green">E175</font></i></a>, <a class="ref" href="sprect_2.html#T16">SPRECT_2:16</a></i>;<br/><a NAME="E113:46_21_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">godo</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K23">W-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E84:46"><i><font color="Green">E166</font></i></a>, <a class="txt" href="jordan15.html#E91:46"><i><font color="Green">E173</font></i></a>, <a class="ref" href="finseq_6.html#T95">FINSEQ_6:95</a></i>;<br/><b>then reconsider </b><font color="Maroon">godo</font> = <font color="Maroon">godo</font> as  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> <b>by </b><i><a class="ref" href="jordan1i.html#T27">JORDAN1I:27</a></i>;<br/><a NAME="E115:46_21_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><b>then </b><i><font color="Green">E256</font></i>: <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1f.html#T7">JORDAN1F:7</a></i>
;<br/><a NAME="E117:46_21_1_1"/><i><font color="Green">E257</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K25">E-most</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T111">PSCOMP_1:111</a></i>;<br/><a NAME="E118:46_21_1_1"/><i><font color="Green">E258</font></i>: 
 <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_22"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_21_1_1_22"/>
 <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46_21_1_1_22"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>
 <b>and </b><br/><a NAME="E4:46_21_1_1_22"/><i><font color="Green">E260</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan15.html#E6:46_21_1_1"><i><font color="Green">E179</font></i></a></i>;<br/>
<a NAME="E6:46_21_1_1_22"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E17:46"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan15.html#E7:46_21_1_1"><i><font color="Green">E180</font></i></a></i>;<br/>
<a NAME="E7:46_21_1_1_22"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K7">east_halfline</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan15.html#E6:46_21_1_1"><i><font color="Green">E179</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:46_21_1_1_22"/><b>then </b><i><font color="Green">E261</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E5:46_21_1_1"><i><font color="Green">E177</font></i></a>, <a class="ref" href="jordan1a.html#T104">JORDAN1A:104</a></i>;<br/>
<a NAME="E9:46_21_1_1_22"/><b>then </b><i><font color="Green">E262</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan15.html#E17:46"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan15.html#E7:46_21_1_1"><i><font color="Green">E180</font></i></a>, <a class="ref" href="jordan1j.html#T46">JORDAN1J:46</a></i>;<br/>
<a NAME="E10:46_21_1_1_22"/><b>then </b>
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <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>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="jordan15.html#E2:46"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan15.html#E4:46_21_1_1"><i><font color="Green">E176</font></i></a>, <a class="txt" href="jordan15.html#E7:46_21_1_1"><i><font color="Green">E180</font></i></a>, <a class="ref" href="jordan1j.html#T43">JORDAN1J:43</a></i>;<br/>
<a NAME="E11:46_21_1_1_22"/><b>then </b>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</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>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, , <a class="txt" href="jordan15.html#E8:46_21_1_1"><i><font color="Green">E181</font></i></a>, <a class="txt" href="jordan15.html#E9:46_21_1_1"><i><font color="Green">E182</font></i></a>, <a class="ref" href="jordan1a.html#T92">JORDAN1A:92</a></i>;<br/>
<b>hence </b><a NAME="E12:46_21_1_1_22"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan15.html#E5:46"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_23"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_21_1_1_23"/>
 <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font>
 ;<br/><a NAME="E2:46_21_1_1_23"/><b>then </b><i><font color="Green">E263</font></i>: 
(  <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R2">meets</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span> or  <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_23_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_21_1_1_23_1"/>
(  <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font> or  <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font> or  <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font> )
 </span><b>by </b><i><a class="txt" href="jordan15.html#E6:46_21_1_1"><i><font color="Green">E179</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_23_1_1"><b>suppose </b></a><a NAME="E1:46_21_1_1_23_1_1"/>
 <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_21_1_1_23_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_23_1_2"><b>suppose </b></a><a NAME="E1:46_21_1_1_23_1_2"/>
 <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46_21_1_1_23_1_2"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>
 <b>and </b><br/><a NAME="E4:46_21_1_1_23_1_2"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan15.html#E7:46_21_1_1"><i><font color="Green">E180</font></i></a></i>;<br/><div><i><font color="Green">E266</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_23_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_23_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="E1:46_21_1_1_23_1_2_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span> )
 </span><b>by </b><i>, <a class="txt" href="jordan15.html#E44:46"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan15.html#E8:46_21_1_1"><i><font color="Green">E181</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_23_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:46_21_1_1_23_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_21_1_1_23_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan15.html#E51:46"><i><font color="Green">E111</font></i></a>, <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_23_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:46_21_1_1_23_1_2_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_21_1_1_23_1_2_1_1_2"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan15.html#E50:46"><i><font color="Green">E110</font></i></a>, <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:46_21_1_1_23_1_2"/>
<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>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:46_21_1_1_23_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:46_21_1_1_23_1_2"/><b>then </b><i><font color="Green">E267</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><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
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E10:46_21_1_1_23_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</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> <a href="binarith.html#K3">-'</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>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E11:46_21_1_1_23_1_2"/><b>then </b>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<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>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="txt" href="jordan15.html#E4:46"><i><font color="Green">E43</font></i></a>, , , <a class="txt" href="jordan15.html#E11:46_21_1_1"><i><font color="Green">E183</font></i></a>, <a class="ref" href="jordan1a.html#T39">JORDAN1A:39</a></i>;<br/><a NAME="E12:46_21_1_1_23_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<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>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan15.html#E9:46_21_1_1"><i><font color="Green">E182</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:46_21_1_1_23_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="jordan8.html#T15">JORDAN8:15</a></i>;<br/><a NAME="E14:46_21_1_1_23_1_2"/><b>then </b><i><font color="Green">E268</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E15:46_21_1_1_23_1_2"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan15.html#E7:46_21_1_1"><i><font color="Green">E180</font></i></a>, <a class="ref" href="topreal1.html#D13">TOPREAL1:def 13</a></i>;<br/><a NAME="E16:46_21_1_1_23_1_2"/><b>then </b><i><font color="Green">E270</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan15.html#E12:46_21_1_1"><i><font color="Green">E184</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E17:46_21_1_1_23_1_2"/>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan15.html#E7:46_21_1_1"><i><font color="Green">E180</font></i></a>, <a class="ref" href="topreal1.html#D13">TOPREAL1:def 13</a></i>;<br/><a NAME="E18:46_21_1_1_23_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E13:46_21_1_1"><i><font color="Green">E185</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/><b>hence </b><a NAME="E19:46_21_1_1_23_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan15.html#E44:46"><i><font color="Green">E106</font></i></a>, , <a class="txt" href="jordan15.html#E8:46_21_1_1"><i><font color="Green">E181</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_23_1_3"><b>suppose </b></a><a NAME="E1:46_21_1_1_23_1_3"/>
 <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46_21_1_1_23_1_3"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>
 <b>and </b><br/><a NAME="E4:46_21_1_1_23_1_3"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan15.html#E7:46_21_1_1"><i><font color="Green">E180</font></i></a></i>;<br/><a NAME="E6:46_21_1_1_23_1_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E19:46"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan15.html#E8:46_21_1_1"><i><font color="Green">E181</font></i></a></i>;<br/><a NAME="E7:46_21_1_1_23_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K7">east_halfline</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan15.html#E7:46_21_1_1"><i><font color="Green">E180</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E273</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E5:46_21_1_1"><i><font color="Green">E177</font></i></a>, <a class="ref" href="jordan1a.html#T104">JORDAN1A:104</a></i>;<br/><a NAME="E9:46_21_1_1_23_1_3"/><i><font color="Green">E274</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan15.html#E7:46_21_1_1"><i><font color="Green">E180</font></i></a>, <a class="ref" href="topreal1.html#D13">TOPREAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon">RC</font> =  <a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E10:46_21_1_1_23_1_3"/><i><font color="Green">E275</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K4">right_cell</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i><a class="ref" href="jordan1i.html#T9">JORDAN1I:9</a></i>;<br/><a NAME="E11:46_21_1_1_23_1_3"/><i><font color="Green">E276</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:46_21_1_1_23_1_3"/>
 <a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_5.html#K1">-:</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan1g.html#T26">JORDAN1G:26</a></i>;<br/><a NAME="E13:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E277</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i><a class="txt" href="jordan15.html#E13:46_21_1_1"><i><font color="Green">E185</font></i></a>, <a class="ref" href="sppol_2.html#T9">SPPOL_2:9</a></i>;<br/><a NAME="E14:46_21_1_1_23_1_3"/><i><font color="Green">E278</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="revrot_1.html#T33">REVROT_1:33</a></i>;<br/><a NAME="E15:46_21_1_1_23_1_3"/><i><font color="Green">E281</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="revrot_1.html#T14">REVROT_1:14</a></i>;<br/><i><font color="Green">E282</font></i>:  <a href="goboard2.html#K3">GoB</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
 <a href="goboard2.html#K3">GoB</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="revrot_1.html#T28">REVROT_1:28</a></i>
<br/>.= 
 <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="jordan1h.html#T52">JORDAN1H:52</a></i>
;<br/><a NAME="E17:46_21_1_1_23_1_3"/><i><font color="Green">E283</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_2.html#T50">SPRECT_2:50</a></i>;<br/><a NAME="E18:46_21_1_1_23_1_3"/><i><font color="Green">E284</font></i>: 
 <a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <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>
 <b>by </b><i><a class="txt" href="jordan15.html#E78:46"><i><font color="Green">E157</font></i></a>, <a class="ref" href="revrot_1.html#T34">REVROT_1:34</a></i>;<br/><a NAME="E19:46_21_1_1_23_1_3"/><i><font color="Green">E286</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E16:46_21_1_1"><i><font color="Green">E187</font></i></a>, <a class="txt" href="jordan15.html#E21:46_21_1_1"><i><font color="Green">E190</font></i></a>, <a class="ref" href="finseq_6.html#T98">FINSEQ_6:98</a></i>;<br/><b>then consider </b><font color="Maroon">ii</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">jj</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E21:46_21_1_1_23_1_3"/><i><font color="Green">E287</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ii</font>,<span class="p2">(<span class="default"><font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <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="E22:46_21_1_1_23_1_3"/><i><font color="Green">E288</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ii</font>,<font color="Maroon">jj</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E23:46_21_1_1_23_1_3"/><i><font color="Green">E289</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <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">ii</font>,<span class="p1">(<span class="default"><font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E24:46_21_1_1_23_1_3"/><i><font color="Green">E290</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">ii</font>,<font color="Maroon">jj</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E58:46"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan15.html#E18:46_21_1_1"><i><font color="Green">E188</font></i></a>, <a class="txt" href="jordan15.html#E22:46_21_1_1"><i><font color="Green">E191</font></i></a>, <a class="ref" href="jordan1i.html#T25">JORDAN1I:25</a></i>;<br/><b>consider </b><font color="Maroon">jj2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E26:46_21_1_1_23_1_3"/><i><font color="Green">E291</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj2</font> &amp; <font color="Maroon">jj2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E27:46_21_1_1_23_1_3"/><i><font color="Green">E292</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <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"><a href="finseq_1.html#K3">len</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>,<font color="Maroon">jj2</font>
 <b>by </b><i><a class="ref" href="jordan1d.html#T29">JORDAN1D:29</a></i>;<br/><a NAME="E28:46_21_1_1_23_1_3"/><i><font color="Green">E293</font></i>: 
 <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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 <b>by </b><i><a class="ref" href="jordan8.html#T13">JORDAN8:13</a></i>;<br/><a NAME="E29:46_21_1_1_23_1_3"/><b>then </b>
 <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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E30:46_21_1_1_23_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</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>,<font color="Maroon">jj2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E29:46_21_1_1"><i><font color="Green">E197</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><a NAME="E31:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E294</font></i>: 
<font color="Maroon">ii</font> <a href="hidden.html#R1">=</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>by </b><i><a class="txt" href="jordan15.html#E16:46_21_1_1"><i><font color="Green">E187</font></i></a>, <a class="txt" href="jordan15.html#E24:46_21_1_1"><i><font color="Green">E192</font></i></a>, <a class="txt" href="jordan15.html#E25:46_21_1_1"><i><font color="Green">E193</font></i></a>, <a class="txt" href="jordan15.html#E27:46_21_1_1"><i><font color="Green">E195</font></i></a>, <a class="txt" href="jordan15.html#E31:46_21_1_1"><i><font color="Green">E198</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E32:46_21_1_1_23_1_3"/><i><font color="Green">E295</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ii</font> &amp; <font color="Maroon">ii</font> <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> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <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="txt" href="jordan15.html#E25:46_21_1_1"><i><font color="Green">E193</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E33:46_21_1_1_23_1_3"/><i><font color="Green">E296</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ii</font> &amp; <font color="Maroon">ii</font> <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> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj</font> &amp; <font color="Maroon">jj</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan15.html#E26:46_21_1_1"><i><font color="Green">E194</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E34:46_21_1_1_23_1_3"/><i><font color="Green">E297</font></i>: 
<font color="Maroon">ii</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ii</font>
 ;<br/><a NAME="E35:46_21_1_1_23_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">jj</font>
 ;<br/><a NAME="E36:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E298</font></i>: 
 <a href="goboard5.html#K4">right_cell</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1 <a href="hidden.html#R1">=</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>,<span class="p1">(<span class="default"><font color="Maroon">ii</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">jj</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E58:46"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan15.html#E18:46_21_1_1"><i><font color="Green">E188</font></i></a>, <a class="txt" href="jordan15.html#E19:46_21_1_1"><i><font color="Green">E189</font></i></a>, <a class="txt" href="jordan15.html#E25:46_21_1_1"><i><font color="Green">E193</font></i></a>, <a class="txt" href="jordan15.html#E26:46_21_1_1"><i><font color="Green">E194</font></i></a>, <a class="txt" href="jordan15.html#E27:46_21_1_1"><i><font color="Green">E195</font></i></a>, <a class="txt" href="jordan15.html#E28:46_21_1_1"><i><font color="Green">E196</font></i></a>, <a class="txt" href="jordan15.html#E39:46_21_1_1"><i><font color="Green">E203</font></i></a>, <a class="ref" href="goboard5.html#D6">GOBOARD5:def 6</a></i>;<br/><a NAME="E37:46_21_1_1_23_1_3"/><i><font color="Green">E299</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">ii</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">ii</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E38:46_21_1_1_23_1_3"/>
<font color="Maroon">ii</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E33:46_21_1_1"><i><font color="Green">E199</font></i></a>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E39:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E300</font></i>: 
<font color="Maroon">ii</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E40:46_21_1_1_23_1_3"/><i><font color="Green">E301</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ii</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E44:46_21_1_1"><i><font color="Green">E206</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E41:46_21_1_1_23_1_3"/><i><font color="Green">E302</font></i>: 
( <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">ii</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">jj</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">ii</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan15.html#E11:46_21_1_1"><i><font color="Green">E183</font></i></a>, <a class="txt" href="jordan15.html#E12:46_21_1_1"><i><font color="Green">E184</font></i></a>, , <a class="txt" href="jordan15.html#E36:46_21_1_1"><i><font color="Green">E202</font></i></a>, <a class="txt" href="jordan15.html#E42:46_21_1_1"><i><font color="Green">E204</font></i></a>, <a class="txt" href="jordan15.html#E43:46_21_1_1"><i><font color="Green">E205</font></i></a>, <a class="txt" href="jordan15.html#E44:46_21_1_1"><i><font color="Green">E206</font></i></a>, <a class="ref" href="jordan9.html#T19">JORDAN9:19</a></i>;<br/><a NAME="E42:46_21_1_1_23_1_3"/><i><font color="Green">E303</font></i>: 
<font color="Maroon">ii</font> <a href="binarith.html#K3">-'</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>by </b><i>, <a class="txt" href="jordan15.html#E43:46_21_1_1"><i><font color="Green">E205</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><i><font color="Green">E304</font></i>: <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">ii</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">jj</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">jj</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E36:46_21_1_1"><i><font color="Green">E202</font></i></a>, <a class="txt" href="jordan15.html#E45:46_21_1_1"><i><font color="Green">E207</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii</font>,<font color="Maroon">jj</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E36:46_21_1_1"><i><font color="Green">E202</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><i><font color="Green">E305</font></i>: <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">ii</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="txt" href="jordan15.html#E45:46_21_1_1"><i><font color="Green">E207</font></i></a>, <a class="txt" href="jordan15.html#E48:46_21_1_1"><i><font color="Green">E209</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii</font>,<span class="p2">(<span class="default"><font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><a NAME="E45:46_21_1_1_23_1_3"/>
( <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</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>,<font color="Maroon">jj</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</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>,<span class="p2">(<span class="default"><font color="Maroon">jj</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , <a class="txt" href="jordan15.html#E36:46_21_1_1"><i><font color="Green">E202</font></i></a>, <a class="ref" href="jordan1a.html#T92">JORDAN1A:92</a></i>;<br/><a NAME="E46:46_21_1_1_23_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E9:46_21_1_1"><i><font color="Green">E182</font></i></a>, <a class="txt" href="jordan15.html#E27:46_21_1_1"><i><font color="Green">E195</font></i></a>, <a class="txt" href="jordan15.html#E28:46_21_1_1"><i><font color="Green">E196</font></i></a>, , <a class="txt" href="jordan15.html#E46:46_21_1_1"><i><font color="Green">E208</font></i></a>, <a class="txt" href="jordan15.html#E50:46_21_1_1"><i><font color="Green">E210</font></i></a>, , <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/><a NAME="E47:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E306</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,1
 <b>by </b><i><a class="txt" href="jordan15.html#E58:46"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan15.html#E14:46_21_1_1"><i><font color="Green">E186</font></i></a>, <a class="txt" href="jordan15.html#E18:46_21_1_1"><i><font color="Green">E188</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E48:46_21_1_1_23_1_3"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">do</font>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E8:46_21_1_1"><i><font color="Green">E181</font></i></a>, <a class="ref" href="jordan3.html#T42">JORDAN3:42</a></i>;<br/><a NAME="E49:46_21_1_1_23_1_3"/><i><font color="Green">E309</font></i>: 
<font color="Maroon">do</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan15.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan1j.html#T37">JORDAN1J:37</a></i>;<br/><a NAME="E50:46_21_1_1_23_1_3"/><i><font color="Green">E310</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan15.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/><a NAME="E51:46_21_1_1_23_1_3"/>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan15.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan15.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>;<br/><a NAME="E52:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E311</font></i>: 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E55:46_21_1_1"><i><font color="Green">E215</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E53:46_21_1_1_23_1_3"/><i><font color="Green">E312</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font> &amp;  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">do</font> )
 <b>by </b><i><a class="txt" href="jordan15.html#E8:46_21_1_1"><i><font color="Green">E181</font></i></a>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/><a NAME="E54:46_21_1_1_23_1_3"/><i><font color="Green">E314</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan15.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan1j.html#T56">JORDAN1J:56</a></i>;<br/><b>consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E56:46_21_1_1_23_1_3"/><i><font color="Green">E315</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E57:46_21_1_1_23_1_3"/><i><font color="Green">E316</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <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>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="jordan15.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E58:46_21_1_1_23_1_3"/><i><font color="Green">E317</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan15.html#E59:46_21_1_1"><i><font color="Green">E219</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E59:46_21_1_1_23_1_3"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="txt" href="jordan15.html#E61:46_21_1_1"><i><font color="Green">E220</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E60:46_21_1_1_23_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E61:46_21_1_1"><i><font color="Green">E220</font></i></a>, <a class="txt" href="jordan15.html#E62:46_21_1_1"><i><font color="Green">E221</font></i></a>, <a class="ref" href="jordan3.html#T45">JORDAN3:45</a></i>;<br/><a NAME="E61:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E318</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E1:46"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan15.html#E61:46_21_1_1"><i><font color="Green">E220</font></i></a>, <a class="ref" href="jordan3.html#T61">JORDAN3:61</a></i>;<br/><b>set </b><font color="Maroon">tt</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E62:46_21_1_1_23_1_3"/><i><font color="Green">E319</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan15.html#E1:46"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/><a NAME="E63:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E320</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E64:46_21_1_1_23_1_3"/>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E58:46_21_1_1"><i><font color="Green">E217</font></i></a>, <a class="txt" href="jordan15.html#E63:46_21_1_1"><i><font color="Green">E222</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E65:46_21_1_1_23_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E66:46_21_1_1_23_1_3"/><b>then </b>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E67:46_21_1_1_23_1_3"/><b>then </b>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E66:46_21_1_1"><i><font color="Green">E224</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E68:46_21_1_1_23_1_3"/><b>then </b>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E69:46_21_1_1_23_1_3"/><b>then </b>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E70:46_21_1_1_23_1_3"/><b>then </b>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E71:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E321</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E55:46_21_1_1"><i><font color="Green">E215</font></i></a>, <a class="txt" href="jordan15.html#E56:46_21_1_1"><i><font color="Green">E216</font></i></a>, <a class="txt" href="jordan15.html#E58:46_21_1_1"><i><font color="Green">E217</font></i></a>, <a class="ref" href="jordan4.html#T31">JORDAN4:31</a></i>;<br/><a NAME="E72:46_21_1_1_23_1_3"/><i><font color="Green">E322</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan15.html#E65:46_21_1_1"><i><font color="Green">E223</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E73:46_21_1_1_23_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E58:46_21_1_1"><i><font color="Green">E217</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E74:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E323</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E75:46_21_1_1_23_1_3"/><b>then </b><i><font color="Green">E324</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E76:46_21_1_1_23_1_3"/><i><font color="Green">E325</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E71:46_21_1_1"><i><font color="Green">E227</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E77:46_21_1_1_23_1_3"/><i><font color="Green">E326</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E13:46_21_1_1"><i><font color="Green">E185</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_23_1_3_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_23_1_3_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_21_1_1_23_1_3_4_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan15.html#E74:46_21_1_1"><i><font color="Green">E229</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_23_1_3_4_1_1"><b>suppose </b></a><a NAME="E1:46_21_1_1_23_1_3_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:46_21_1_1_23_1_3_4_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,1 <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><b>hence </b><a NAME="E3:46_21_1_1_23_1_3_4_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan15.html#E53:46_21_1_1"><i><font color="Green">E213</font></i></a>, <a class="txt" href="jordan15.html#E54:46_21_1_1"><i><font color="Green">E214</font></i></a>, <a class="txt" href="jordan15.html#E69:46_21_1_1"><i><font color="Green">E225</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_23_1_3_4_1_2"><b>suppose </b></a><a NAME="E1:46_21_1_1_23_1_3_4_1_2"/><i><font color="Green">E328</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:46_21_1_1_23_1_3_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,1</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><a NAME="E3:46_21_1_1_23_1_3_4_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i>, <a class="txt" href="jordan15.html#E53:46_21_1_1"><i><font color="Green">E213</font></i></a>, <a class="txt" href="jordan15.html#E54:46_21_1_1"><i><font color="Green">E214</font></i></a>, <a class="txt" href="jordan15.html#E69:46_21_1_1"><i><font color="Green">E225</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:46_21_1_1_23_1_3_4_1_2"/><b>then </b><i><font color="Green">E329</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:46_21_1_1_23_1_3_4_1_2"/><b>then </b><i><font color="Green">E330</font></i>: 
<font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="txt" href="jordan15.html#E75:46_21_1_1"><i><font color="Green">E230</font></i></a>, <a class="ref" href="finseq_5.html#T44">FINSEQ_5:44</a></i>;<br/><a NAME="E6:46_21_1_1_23_1_3_4_1_2"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan15.html#E73:46_21_1_1"><i><font color="Green">E228</font></i></a>, <a class="txt" href="jordan15.html#E77:46_21_1_1"><i><font color="Green">E231</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E7:46_21_1_1_23_1_3_4_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">do</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/><a NAME="E8:46_21_1_1_23_1_3_4_1_2"/><b>then </b><i><font color="Green">E331</font></i>: 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="txt" href="jordan15.html#E58:46_21_1_1"><i><font color="Green">E217</font></i></a>, <a class="txt" href="jordan15.html#E70:46_21_1_1"><i><font color="Green">E226</font></i></a>, <a class="ref" href="jordan1e.html#T10">JORDAN1E:10</a></i>;<br/><a NAME="E9:46_21_1_1_23_1_3_4_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E75:46_21_1_1"><i><font color="Green">E230</font></i></a>, <a class="txt" href="jordan15.html#E79:46_21_1_1"><i><font color="Green">E232</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E10:46_21_1_1_23_1_3_4_1_2"/><b>then </b>
<font color="Maroon">p</font> <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>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="jordan15.html#T11" target="_self">Th11</a>, <a class="txt" href="jordan15.html#E81:46_21_1_1"><i><font color="Green">E233</font></i></a>, <a class="txt" href="jordan15.html#E82:46_21_1_1"><i><font color="Green">E234</font></i></a>, <a class="ref" href="finseq_5.html#T10">FINSEQ_5:10</a></i>;<br/><a NAME="E11:46_21_1_1_23_1_3_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E79:46_21_1_1"><i><font color="Green">E232</font></i></a>, <a class="ref" href="jordan1g.html#T40">JORDAN1G:40</a></i>;<br/><a NAME="E12:46_21_1_1_23_1_3_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</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>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan15.html#E3:46"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="jordan1a.html#T92">JORDAN1A:92</a></i>;<br/><b>hence </b><a NAME="E13:46_21_1_1_23_1_3_4_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan15.html#E32:46"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E79:46_21_1_1_23_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E120:46_21_1_1"/><b>then </b>
 <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E122:46_21_1_1"/><i><font color="Green">E332</font></i>: 
<font color="Maroon">W</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>and </b><br/><a NAME="E123:46_21_1_1"/><i><font color="Green">E353</font></i>: 
 <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="goboard9.html#T5">GOBOARD9:5</a></i>;<br/><a NAME="E124:46_21_1_1"/>
not  <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T129">JORDAN2C:129</a></i>;<br/><a NAME="E125:46_21_1_1"/><b>then </b>
not <font color="Maroon">W</font> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i><a class="txt" href="jordan15.html#E7:46_21_1_1"><i><font color="Green">E180</font></i></a>, <a class="ref" href="jordan2c.html#T16">JORDAN2C:16</a></i>;<br/><a NAME="E126:46_21_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E6:46_21_1_1"><i><font color="Green">E179</font></i></a>, <a class="ref" href="jordan2c.html#D4">JORDAN2C:def 4</a></i>;<br/><a NAME="E127:46_21_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T27">JORDAN2C:27</a></i>;<br/><a NAME="E128:46_21_1_1"/><b>then </b><i><font color="Green">E354</font></i>: 
 <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E7:46_21_1_1"><i><font color="Green">E180</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E129:46_21_1_1"/>
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T45">TOPREAL1:45</a></i>;<br/><a NAME="E130:46_21_1_1"/><b>then </b>
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E8:46_21_1_1"><i><font color="Green">E181</font></i></a></i>;<br/><a NAME="E131:46_21_1_1"/><b>then </b>
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">godo</font>
 <b>by </b><i><a class="ref" href="gobrd14.html#T46">GOBRD14:46</a></i>;<br/><a NAME="E132:46_21_1_1"/><b>then </b>
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">godo</font>
 <b>by </b><i>, , , <a class="txt" href="jordan15.html#E83:46"><i><font color="Green">E165</font></i></a>, <a class="ref" href="jordan1j.html#T36">JORDAN1J:36</a></i>;<br/><a NAME="E133:46_21_1_1"/><b>then </b><i><font color="Green">E355</font></i>: 
(  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font></span>)</span> or  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><a NAME="E134:46_21_1_1"/><i><font color="Green">E356</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T76">JORDAN6:76</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_21_1_1_24"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_21_1_1_24_1"/>
(  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font> or  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font> or  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font> )
 </span><b>by </b><i><a class="txt" href="jordan15.html#E9:46_21_1_1"><i><font color="Green">E182</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_24_1_1"><b>suppose </b></a><a NAME="E1:46_21_1_1_24_1_1"/>
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">go</font>
 ;<br/><div class="add"><a NAME="E2:46_21_1_1_24_1_1"/><b>then </b>
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E3:46_21_1_1_24_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E11:46_21_1_1"><i><font color="Green">E183</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><b>hence </b><a NAME="E4:46_21_1_1_24_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan10.html#T5">JORDAN10:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_24_1_2"><b>suppose </b></a><a NAME="E1:46_21_1_1_24_1_2"/>
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">pion1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_21_1_1_24_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan15.html#E44:46"><i><font color="Green">E106</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_1_24_1_3"><b>suppose </b></a><a NAME="E1:46_21_1_1_24_1_3"/>
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">do</font>
 ;<br/><div class="add"><a NAME="E2:46_21_1_1_24_1_3"/><b>then </b>
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E3:46_21_1_1_24_1_3"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E11:46_21_1_1"><i><font color="Green">E183</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><b>hence </b><a NAME="E4:46_21_1_1_24_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan10.html#T5">JORDAN10:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E136:46_21_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_2"><b>suppose </b></a><a NAME="E1:46_21_1_2"/>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:46_21_1_2"/><b>then </b><i><font color="Green">E357</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="jordan15.html#E5:46"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan1g.html#T7">JORDAN1G:7</a></i>;<br/><a NAME="E3:46_21_1_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T7">TOPREAL1:7</a></i>;<br/><a NAME="E4:46_21_1_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E41:46"><i><font color="Green">E102</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E5:46_21_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>hence </b><a NAME="E6:46_21_1_2"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="jordan15.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan1j.html#T59">JORDAN1J:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_21_1_3"><b>suppose </b></a><a NAME="E1:46_21_1_3"/>
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:46_21_1_3"/><b>then </b><i><font color="Green">E358</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="jordan15.html#E5:46"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan1g.html#T6">JORDAN1G:6</a></i>;<br/><a NAME="E3:46_21_1_3"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T7">TOPREAL1:7</a></i>;<br/><a NAME="E4:46_21_1_3"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E42:46"><i><font color="Green">E103</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E5:46_21_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>hence </b><a NAME="E6:46_21_1_3"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="jordan15.html#T13" target="_self">Th13</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E95:46"/>
contradiction
 ;<br/>


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