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

<b>set </b><font color="Maroon">m</font> =  <a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 holds <br/> <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"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Olive">k</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font> ) );<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] </span>}</span> ;<br/>
<a NAME="E1:8_1_1"/><i><font color="Green">E45</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:8_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <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>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">j0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/>
<a NAME="E3:8_1_1"/>
<span class="p1">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</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">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan1a.html#T49">JORDAN1A:49</a></i>;<br/>
<a NAME="E4:8_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</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"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E5:8_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:8_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:8_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="sprect_3.html#T7">SPRECT_3:7</a></i>;<br/>
<a NAME="E8:8_1_1"/>
<span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E9:8_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/>
<a NAME="E10:8_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E11:8_1_1"/>
2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E12:8_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <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="E13:8_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:8_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
;<br/>
<a NAME="E15:8_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <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="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E16:8_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <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="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:8_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 ;<br/><a NAME="E2:8_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:8_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T19">PREPOWER:19</a></i>;<br/><a NAME="E4:8_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
2 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><i><font color="Green">E56</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>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1

;<br/><a NAME="E6:8_1_1_1"/>
<span class="p1">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E7:8_1_1_1"/><b>then </b>
2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T107">PREPOWER:107</a></i>;<br/><a NAME="E8:8_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:8_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</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="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</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="E10:8_1_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i/>;<br/><a NAME="E11:8_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <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/><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1

<br/>.= 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><a NAME="E13:8_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2
 <b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/><a NAME="E14:8_1_1_1"/>
 <a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</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="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E15:8_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E16:8_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <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"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</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>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , , , , , <a class="ref" href="jordan1a.html#T69">JORDAN1A:69</a></i>;<br/><a NAME="E17:8_1_1_1"/>
 <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>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E18:8_1_1_1"/>
 <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"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:8_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E19:8_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>take </b>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">X</font>
;<br/>

<a NAME="E21:8_1_1"/>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E22:8_1_1"/><b>then </b>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] )
 
;<br/>
<b>hence </b><a NAME="E23:8_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">X</font>]
 ;<br/>

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

<b>assume </b><a NAME="E24:8_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>]
 ;<br/>

<a NAME="E25:8_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<b>hence </b><a NAME="E26:8_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>


</div>
