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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font> being <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c1">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Olive" title="b1">C</font> holds <br/> <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Olive" title="b1">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Olive" title="b1">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span><br/><b>let </b><font color="Maroon" title="c2">C</font> be  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c2">C</font> implies  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )</span><br/>



<b>set </b><font color="Maroon" title="c3">m</font> =  <a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font>;<br/>
<a NAME="E1:10"/><i><font color="Green" title="E11">A1</font></i>: 
 <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<span class="p1">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 
<b>by </b><i><a class="ref" href="jordan1h.html#T58" title="JORDAN1H:th.58">JORDAN1H:58</a></i>;<br/>
<b>assume </b><a NAME="E2:10"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c2">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span><br/>

<a NAME="E3:10"/><b>then </b><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="jordan11.html#D1" target="_self" title="JORDAN11:def.1">Def1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">i1</font> =  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>;<br/>
<b>set </b><font color="Maroon" title="c5">j0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2;<br/>
<a NAME="E4:10"/><i><font color="Green" title="E14">A4</font></i>: 
<span class="p1">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan11.html#T3" target="_self" title="JORDAN11:th.3">Th3</a></i>;<br/>
<a NAME="E5:10"/><i><font color="Green" title="E15">A5</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="jordan11.html#T2" target="_self" title="JORDAN11:th.2">Th2</a></i>;<br/>
<a NAME="E6:10"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:10"/><b>then </b><i><font color="Green" title="E16">A6</font></i>: 
<span class="p1">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/>
<a NAME="E8:10"/>
 <a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E9:10"/><b>then </b>
 <a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan11.html#T3" target="_self" title="JORDAN11:th.3">Th3</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:10"/><b>then </b><i><font color="Green" title="E17">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:10"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="#E5:10"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="#E7:10"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="jordan1a.html#T53" title="JORDAN1A:th.53">JORDAN1A:53</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 implies  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c6">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">C</font> )</span><br/><b>assume </b><a NAME="E1:10_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c6">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">C</font></span><br/><a NAME="E2:10_1"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:10_1"/><i><font color="Green" title="E19">A9</font></i>: 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<span class="p3">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="ref" href="jordan11.html#D2" target="_self" title="JORDAN11:def.2">Def2</a></i>;<br/><a NAME="E4:10_1"/><i><font color="Green" title="E20">A10</font></i>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<span class="p1">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:10_1"/><i><font color="Green" title="E21">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 <b>by </b><i><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></i>;<br/><a NAME="E6:10_1"/>
<span class="p1">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></i>;<br/><a NAME="E7:10_1"/><b>then </b>
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T107" title="PREPOWER:th.107">PREPOWER:107</a></i>;<br/><a NAME="E8:10_1"/><b>then </b><i><font color="Green" title="E22">A12</font></i>: 
 <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<span class="p1">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:10_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E9:10_1"/>
 <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="jordan11.html#T4" target="_self" title="JORDAN11:th.4">Th4</a></i>;<br/><a NAME="E10:10_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c6">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<span class="p3">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:10"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="#E4:10"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="#E7:10"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="#E2:10_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="#E4:10_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="#E8:10_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="jordan11.html#T2" target="_self" title="JORDAN11:th.2">Th2</a>, <a class="ref" href="jordan1a.html#T69" title="JORDAN1A:th.69">JORDAN1A:69</a></i>;<br/><b>hence </b><a NAME="E11:10_1"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c6">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="#E3:10_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:10"/>
 <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E10:10"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="jordan11.html#D3" target="_self" title="JORDAN11:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
