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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="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="Olive" title="b1">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="Olive" title="b1">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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">C</font></span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">C</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">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/>
<a NAME="E1:15"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<span class="p1">(<span class="default"><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>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T146" onmouseover="rs('xreal_1/T146')" onmouseout="rh()">XREAL_1:146</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:15"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">C</font></span><br/>

<a NAME="E3:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A3</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan11.html#T7" target="_self" onmouseover="rs('jordan11/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:15"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 <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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">C</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:15"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1">(<span class="default"><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>)</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> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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#K11" title="NEWTON:func.11">|^</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#K2" title="NAT_1:func.2">*</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#K1" title="NAT_1:func.1">+</a> 2 holds <br/> <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="Olive" title="b1">k</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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><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>)</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> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c4">k</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#K11" title="NEWTON:func.11">|^</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#K2" title="NAT_1:func.2">*</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#K1" title="NAT_1:func.1">+</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="c4">k</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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15_1"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<span class="p1">(<span class="default"><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>)</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> <font color="Maroon" title="c4">k</font>
 <span class="kw">and </span><br/><a NAME="E2:15_1"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<font color="Maroon" title="c4">k</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#K11" title="NEWTON:func.11">|^</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#K2" title="NAT_1:func.2">*</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#K1" title="NAT_1:func.1">+</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c4">k</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_1_1"/>
( <span class="p1">(<span class="default"><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>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> or <span class="p1">(<span class="default"><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>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_1"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<span class="p1">(<span class="default"><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>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c4">k</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/><div class="add"><a NAME="E2:15_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="jordan11.html#T7" target="_self" onmouseover="rs('jordan11/T7')" onmouseout="rh()">Th7</a></span>;<br/><a NAME="E3:15_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="xreal_1.html#T235" onmouseover="rs('xreal_1/T235')" onmouseout="rh()">XREAL_1:235</a></span>;<br/><a NAME="E4:15_1_1_1"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
 <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="c4">k</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E1:15_1_1_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="subset_1.html#T23" onmouseover="rs('subset_1/T23')" onmouseout="rh()">SUBSET_1:23</a></span>;<br/><a NAME="E5:15_1_1_1"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E6:15_1_1_1"/>
 <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>  <a href="matrix_0.html#K1" title="MATRIX_0: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="jordan11.html#T7" target="_self" onmouseover="rs('jordan11/T7')" onmouseout="rh()">Th7</a></span>;<br/><a NAME="E7:15_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A12</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_0.html#K1" title="MATRIX_0: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E5:15_1_1_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">W</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <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> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">C</font> </span> } </span> ;<br/><a NAME="E8:15_1_1_1"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<span class="p1">(<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 <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>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" onmouseover="rs('nat_d/T44')" onmouseout="rh()">NAT_D:44</a></span>;<br/><a NAME="E9:15_1_1_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="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>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T49" onmouseover="rs('jordan1h/T49')" onmouseout="rh()">JORDAN1H:49</a></span>;<br/><a NAME="E10:15_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<span class="p1">(<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 <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>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1_1_1"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E11:15_1_1_1"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <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> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">C</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D4" onmouseover="rs('jordan2c/D4')" onmouseout="rh()">JORDAN2C:def 4</a></span>;<br/><a NAME="E12:15_1_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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 class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T49" onmouseover="rs('jordan1h/T49')" onmouseout="rh()">JORDAN1H:49</a></span>;<br/><a NAME="E13:15_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E14:15_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<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 class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T39" onmouseover="rs('nat_d/T39')" onmouseout="rh()">NAT_D:39</a></span>;<br/><a NAME="E15:15_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p2">(<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="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>,<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="c4">k</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p2">(<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="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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><span class="p2">(<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> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<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> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15_1_1_1"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E7:15_1_1_1"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="goboard5.html#T26" onmouseover="rs('goboard5/T26')" onmouseout="rh()">GOBOARD5:26</a></span>;<br/><a NAME="E16:15_1_1_1"/><span class="kw">then </span>
 <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="c4">k</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</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>,<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>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 ;<br/><a NAME="E17:15_1_1_1"/><span class="kw">then </span>
 <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="c4">k</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:15_1_1_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="jordan11.html#T6" target="_self" onmouseover="rs('jordan11/T6')" onmouseout="rh()">Th6</a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E19:15_1_1_1"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <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> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">C</font> </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E20:15_1_1_1"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
 <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="c4">k</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1_1"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T80" onmouseover="rs('zfmisc_1/T80')" onmouseout="rh()">ZFMISC_1:80</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">B</font> being   <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><span class="kw"> such that </span><br/><a NAME="E22:15_1_1_1"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">and </span><br/><a NAME="E23:15_1_1_1"/><span class="lab"><font color="Green" title="E31">A19</font></span>: 
<font color="Maroon" title="c7">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:15_1_1_1"><span class="lab"><font color="Green" title="E28">A16</font></span></a></span>;<br/><a NAME="E24:15_1_1_1"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
<font color="Maroon" title="c7">B</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1_1"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E19:15_1_1_1"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E22:15_1_1_1"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><a NAME="E25:15_1_1_1"/>
<font color="Maroon" title="c7">B</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15_1_1_1"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="jordan2c.html#D2" onmouseover="rs('jordan2c/D2')" onmouseout="rh()">JORDAN2C:def 2</a></span>;<br/><a NAME="E26:15_1_1_1"/><span class="kw">then </span>
 <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="c4">k</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_1_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E10:15_1_1_1"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E7:15_1_1_1"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="txt" href="#E20:15_1_1_1"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="txt" href="#E22:15_1_1_1"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="goboard9.html#T4" onmouseover="rs('goboard9/T4')" onmouseout="rh()">GOBOARD9:4</a>, <a class="ref" href="jordan1a.html#T25" onmouseover="rs('jordan1a/T25')" onmouseout="rh()">JORDAN1A:25</a></span>;<br/><span class="kw">hence </span><a NAME="E27:15_1_1_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="c4">k</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E24:15_1_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_2"/>
<span class="p1">(<span class="default"><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>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c4">k</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/><div class="add"><a NAME="E2:15_1_1_2"/><span class="kw">then </span>
 <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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T55" onmouseover="rs('nat_d/T55')" onmouseout="rh()">NAT_D:55</a></span>;<br/><a NAME="E3:15_1_1_2"/><span class="kw">then </span>
 <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> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:15_1_1_2"/>
 <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="c4">k</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="jordan11.html#D3" target="_self" onmouseover="rs('jordan11/D3')" onmouseout="rh()">Def3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E6:15"/>
 <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>  <a href="matrix_0.html#K1" title="MATRIX_0: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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="jordan11.html#D3" target="_self" onmouseover="rs('jordan11/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E7:15"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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>)</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="matrix_0.html#K1" title="MATRIX_0: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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" onmouseover="rs('nat_d/T44')" onmouseout="rh()">NAT_D:44</a></span>;<br/>
<a NAME="E8:15"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="jordan11.html#D3" target="_self" onmouseover="rs('jordan11/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:15"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
