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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">k</font> being    <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>   st <font color="Olive" title="b1">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">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#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">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="c1">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="c1">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 holds <br/><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="c1">C</font>,<font color="Olive" title="b1">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="c1">C</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b2">k</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font>)</span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">k</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">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#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">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="c1">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="c1">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 <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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">AI</font> =  <a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">YI</font> =  <a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">XS</font> =  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">YS</font> =  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:33"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">and </span><br/><a NAME="E3:33"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c3">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#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">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="c1">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="c1">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span></span><br/>

<a NAME="E4:33"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
 <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">ro</font> = <font color="Maroon" title="c3">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/>
<a NAME="E6:33"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon" title="c10">ro</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"><span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">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>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E7:33"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">ro</font>
 
;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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> ] <span class="kw">means </span>( $1 <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"><span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">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>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> implies <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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> );<br/>
<a NAME="E8:33"/><span class="lab"><font color="Green" title="E32">A7</font></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="c1">C</font>,<font color="Maroon" title="c2">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="jordan1h.html#T50" title="JORDAN1H:th.50">JORDAN1H:50</a></span>;<br/>
<a NAME="E9:33"/>
 <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T49" title="JORDAN1H:th.49">JORDAN1H:49</a></span>;<br/>
<a NAME="E10:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A8</font></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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:33"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T50" title="JORDAN1H:th.50">JORDAN1H:50</a></span>;<br/>
<a NAME="E12:33"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
 for <font color="Olive" title="b1">t</font> being    <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">t</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">t</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
( <font color="Maroon" title="c11">t</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"><span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">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>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> implies <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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">t</font></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">Ls</font> =  <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1: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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>);<br/>
<a NAME="E2:33_1"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c11">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">p</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span></span>)</span>;<br/>
<a NAME="E3:33_1"/><span class="lab"><font color="Green" title="E38">A13</font></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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> or <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_1"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/>

<a NAME="E2:33_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:33_1_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="subset_1.html#T29" title="SUBSET_1:th.29">SUBSET_1:29</a></span>; <a class="txt" 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>
<a NAME="E4:33_1"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1: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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T69" title="RLTOPSP1:th.69">RLTOPSP1:69</a></span>;<br/>
<a NAME="E5:33_1"/><span class="lab"><font color="Green" title="E40">A16</font></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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E6:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A17</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E7:33_1"/><span class="lab"><font color="Green" title="E42">A18</font></span>: 
 <a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan11.html#T2" title="JORDAN11:th.2">JORDAN11:2</a></span>;<br/>
<a NAME="E8:33_1"/><span class="kw">then </span>
 <a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E9:33_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E10:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A19</font></span>: 
<span class="p1">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">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="c1">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:33_1"/><span class="lab"><font color="Green" title="E44">A20</font></span>: 
<font color="Maroon" title="c11">t</font> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">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>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" 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="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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span></span><br/>

<a NAME="E12:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A21</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <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#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">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="c1">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="c1">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 class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<span class="kw">assume </span><a NAME="E13:33_1"/>
 not <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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:33_1"/><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:33_1"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
 not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E17:33_1"/>
 not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:33_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E18:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:33_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="subset_1.html#T29" title="SUBSET_1:th.29">SUBSET_1:29</a></span>;<br/>
<a NAME="E19:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T27" title="JORDAN2C:th.27">JORDAN2C:27</a></span>;<br/>
<a NAME="E20:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:33_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E21:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A24</font></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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:33_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E22:33_1"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
 <a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">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="c1">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan11.html#D2" title="JORDAN11:def.2">JORDAN11:def 2</a></span>;<br/>
<a NAME="E23:33_1"/>
 <a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="jordan11.html#D1" title="JORDAN11:def.1">JORDAN11:def 1</a></span>;<br/>
<a NAME="E24:33_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">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="c1">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="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">C</font></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 <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E22:33_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="ref" href="jordan1a.html#T32" title="JORDAN1A:th.32">JORDAN1A:32</a></span>;<br/>
<a NAME="E25:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A26</font></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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">t</font></span>)</span> <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="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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E26:33_1"/><span class="lab"><font color="Green" title="E51">A27</font></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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">t</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">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="c1">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="c1">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E52">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:33_1_2"/><span class="lab"><font color="Green" title="E53">A29</font></span>: 
 <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E2:33_1_2"/><span class="lab"><font color="Green" title="E54">A30</font></span>: 
 <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T49" title="JORDAN1H:th.49">JORDAN1H:49</a></span>;<br/><span class="kw">assume </span><a NAME="E3:33_1_2"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">i</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E5:33_1_2"/><span class="lab"><font color="Green" title="E55">A31</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">and </span><br/><a NAME="E6:33_1_2"/><span class="lab"><font color="Green" title="E56">A32</font></span>: 
<font color="Maroon" title="c15">i</font> <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:33_1_2"/><span class="lab"><font color="Green" title="E57">A33</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c15">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T13" title="SPPOL_2:th.13">SPPOL_2:13</a></span>;<br/><a NAME="E8:33_1_2"/><span class="lab"><font color="Green" title="E58">A34</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c15">i</font>) <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E6:33_1_2"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">i1</font>, <font color="Maroon" title="c17">j1</font>, <font color="Maroon" title="c18">i2</font>, <font color="Maroon" title="c19">j2</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E10:33_1_2"/><span class="lab"><font color="Green" title="E59">A35</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:33_1_2"/><span class="lab"><font color="Green" title="E60">A36</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font>)
 <span class="kw">and </span><br/><a NAME="E12:33_1_2"/><span class="lab"><font color="Green" title="E61">A37</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2</font>,<font color="Maroon" title="c19">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:33_1_2"/><span class="lab"><font color="Green" title="E62">A38</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i2</font>,<font color="Maroon" title="c19">j2</font>)
 <span class="kw">and </span><br/><a NAME="E14:33_1_2"/><span class="lab"><font color="Green" title="E63">A39</font></span>: 
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E6:33_1_2"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E15:33_1_2"/><span class="lab"><font color="Green" title="E64">A40</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_1_2"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E16:33_1_2"/><span class="lab"><font color="Green" title="E65">A41</font></span>: 
<font color="Maroon" title="c18">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1_2"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E17:33_1_2"/><span class="lab"><font color="Green" title="E66">A42</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1_2"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E18:33_1_2"/><span class="lab"><font color="Green" title="E67">A43</font></span>: 
<font color="Maroon" title="c17">j1</font> <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_1_2"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E19:33_1_2"/><span class="lab"><font color="Green" title="E68">A44</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1_2"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E20:33_1_2"/><span class="lab"><font color="Green" title="E69">A45</font></span>: 
<font color="Maroon" title="c16">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_1_2"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E21:33_1_2"/><span class="lab"><font color="Green" title="E70">A46</font></span>: 
<font color="Maroon" title="c19">j2</font> <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1_2"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E22:33_1_2"/><span class="lab"><font color="Green" title="E71">A47</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_1_2"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:33_1_2_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:33_1_2"><span class="lab"><font color="Green" title="E63">A39</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:33_1_2_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:33_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E25:33_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E7:33_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E8:33_1_2"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E11:33_1_2"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E13:33_1_2"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E15:33_1_2"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="txt" href="#E20:33_1_2"><span class="lab"><font color="Green" title="E69">A45</font></span></a>, <a class="txt" href="#E22:33_1_2"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E21:33_1_2"><span class="lab"><font color="Green" title="E70">A46</font></span></a>, <a class="txt" href="#E2:33_1_2"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="goboard7.html#T27" title="GOBOARD7:th.27">GOBOARD7:27</a></span>; <a class="txt" 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="33_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:33_1_2_1_2"/><span class="lab"><font color="Green" title="E72">A48</font></span>: 
( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:33_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A49</font></span>: 
<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E25:33_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E7:33_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E8:33_1_2"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E11:33_1_2"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E13:33_1_2"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E15:33_1_2"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="txt" href="#E22:33_1_2"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E18:33_1_2"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="txt" href="#E16:33_1_2"><span class="lab"><font color="Green" title="E65">A41</font></span></a>, <a class="txt" href="#E2:33_1_2"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="goboard7.html#T26" title="GOBOARD7:th.26">GOBOARD7:26</a></span>;<br/><a NAME="E3:33_1_2_1_2"/><span class="lab"><font color="Green" title="E74">A50</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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E1:33_1_2"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="jordan11.html#D3" title="JORDAN11:def.3">JORDAN11:def 3</a></span>;<br/><a NAME="E4:33_1_2_1_2"/><span class="lab"><font color="Green" title="E75">A51</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c15">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E6:33_1_2"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E10:33_1_2"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E11:33_1_2"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E12:33_1_2"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="txt" href="#E13:33_1_2"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E1:33_1_2_1_2"><span class="lab"><font color="Green" title="E72">A48</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><a NAME="E5:33_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="c16">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E25:33_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E7:33_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E8:33_1_2"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E11:33_1_2"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E13:33_1_2"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E15:33_1_2"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="txt" href="#E22:33_1_2"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E18:33_1_2"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="txt" href="#E16:33_1_2"><span class="lab"><font color="Green" title="E65">A41</font></span></a>, <a class="txt" href="#E2:33_1_2"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E1:33_1_2_1_2"><span class="lab"><font color="Green" title="E72">A48</font></span></a>, <a class="ref" href="goboard7.html#T26" title="GOBOARD7:th.26">GOBOARD7:26</a></span>;<br/><a NAME="E6:33_1_2_1_2"/><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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E6:33_1_2"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E2:33_1_2_1_2"><span class="lab"><font color="Green" title="E73">A49</font></span></a>, <a class="txt" href="#E4:33_1_2_1_2"><span class="lab"><font color="Green" title="E75">A51</font></span></a>, <a class="ref" href="jordan14.html#T7" target="_self" title="JORDAN14:th.7">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E7:33_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_2_1_2"><span class="lab"><font color="Green" title="E74">A50</font></span></a>, <a class="ref" href="jordan1a.html#T7" title="JORDAN1A:th.7">JORDAN1A:7</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" 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="33_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:33_1_2_1_3"/><span class="lab"><font color="Green" title="E72">A52</font></span>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:33_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A53</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c15">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c18">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E6:33_1_2"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E10:33_1_2"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E11:33_1_2"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E12:33_1_2"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="txt" href="#E13:33_1_2"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><a NAME="E3:33_1_2_1_3"/><span class="lab"><font color="Green" title="E74">A54</font></span>: 
<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E25:33_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E7:33_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E8:33_1_2"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E11:33_1_2"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E13:33_1_2"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E20:33_1_2"><span class="lab"><font color="Green" title="E69">A45</font></span></a>, <a class="txt" href="#E22:33_1_2"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E18:33_1_2"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="txt" href="#E17:33_1_2"><span class="lab"><font color="Green" title="E66">A42</font></span></a>, <a class="txt" href="#E2:33_1_2"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E1:33_1_2_1_3"><span class="lab"><font color="Green" title="E72">A52</font></span></a>, <a class="ref" href="goboard7.html#T26" title="GOBOARD7:th.26">GOBOARD7:26</a></span>;<br/><a NAME="E4:33_1_2_1_3"/>
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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="c18">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E25:33_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E7:33_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E8:33_1_2"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E11:33_1_2"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E13:33_1_2"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E20:33_1_2"><span class="lab"><font color="Green" title="E69">A45</font></span></a>, <a class="txt" href="#E22:33_1_2"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E18:33_1_2"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="txt" href="#E17:33_1_2"><span class="lab"><font color="Green" title="E66">A42</font></span></a>, <a class="txt" href="#E2:33_1_2"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E1:33_1_2_1_3"><span class="lab"><font color="Green" title="E72">A52</font></span></a>, <a class="ref" href="goboard7.html#T26" title="GOBOARD7:th.26">GOBOARD7:26</a></span>;<br/><a NAME="E5:33_1_2_1_3"/><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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E6:33_1_2"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E3:33_1_2_1_3"><span class="lab"><font color="Green" title="E74">A54</font></span></a>, <a class="txt" href="#E2:33_1_2_1_3"><span class="lab"><font color="Green" title="E73">A53</font></span></a>, <a class="ref" href="jordan14.html#T7" target="_self" title="JORDAN14:th.7">Th8</a></span>;<br/><a NAME="E6:33_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A55</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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">t</font></span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E7:33_1_2_1_3"/><span class="lab"><font color="Green" title="E76">A56</font></span>: 
 <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E8:33_1_2_1_3"/>
<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">t</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#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">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="c1">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="c1">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E26:33_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E9:33_1_2_1_3"/><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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">t</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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E7:33_1_2_1_3"><span class="lab"><font color="Green" title="E76">A56</font></span></a>, <a class="ref" href="jordan11.html#D3" title="JORDAN11:def.3">JORDAN11:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:33_1_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1_2_1_3"><span class="lab"><font color="Green" title="E75">A55</font></span></a>, <a class="ref" href="jordan1a.html#T7" title="JORDAN1A:th.7">JORDAN1A:7</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" 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="33_1_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:33_1_2_1_4"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:33_1_2_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E25:33_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E7:33_1_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E8:33_1_2"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E11:33_1_2"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E13:33_1_2"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E15:33_1_2"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="txt" href="#E20:33_1_2"><span class="lab"><font color="Green" title="E69">A45</font></span></a>, <a class="txt" href="#E18:33_1_2"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="txt" href="#E19:33_1_2"><span class="lab"><font color="Green" title="E68">A44</font></span></a>, <a class="txt" href="#E2:33_1_2"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="goboard7.html#T27" title="GOBOARD7:th.27">GOBOARD7:27</a></span>; <a class="txt" 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="E28:33_1"/>
<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">t</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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:33_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E29:33_1"/><span class="kw">then </span>
 <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1: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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>) <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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">t</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E11:33"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="goboard5.html#T21" title="GOBOARD5:th.21">GOBOARD5:21</a></span>;<br/>
<a NAME="E30:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A57</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">t</font></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:33_1"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E31:33_1"/>
 <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1: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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span>) <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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E11:33"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E25:33_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="goboard5.html#T22" title="GOBOARD5:th.22">GOBOARD5:22</a></span>;<br/>
<a NAME="E32:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A58</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:33_1"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E33:33_1"/>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></span>;<br/>
<a NAME="E34:33_1"/><span class="kw">then </span>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T36" title="GOBRD14:th.36">GOBRD14:36</a></span>;<br/>
<a NAME="E35:33_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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E11:33"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E25:33_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E21:33_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="jordan14.html#T29" target="_self" title="JORDAN14:th.29">Th30</a>, <a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a></span>;<br/>
<a NAME="E36:33_1"/><span class="kw">then </span>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E11:33_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E30:33_1"><span class="lab"><font color="Green" title="E53">A57</font></span></a>, <a class="txt" href="#E32:33_1"><span class="lab"><font color="Green" title="E54">A58</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E37:33_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T24" title="JORDAN2C:th.24">JORDAN2C:24</a></span>; <a class="txt" 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>
<a NAME="E13:33"/><span class="lab"><font color="Green" title="E36">A59</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:33_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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"><span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p5">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan11.html#K2" title="JORDAN11:func.2">Y-InitStart</a> <font color="Maroon" title="c1">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>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" 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="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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span></span><br/>

<a NAME="E2:33_2"/><span class="lab"><font color="Green" title="E37">A60</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<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="c1">C</font>,<font color="Maroon" title="c2">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"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></span>;<br/>
<a NAME="E3:33_2"/><span class="lab"><font color="Green" title="E38">A61</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="jordan11.html#T8" title="JORDAN11:th.8">JORDAN11:8</a></span>;<br/>
<a NAME="E4:33_2"/><span class="lab"><font color="Green" title="E39">A62</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><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="c1">C</font>,<font color="Maroon" title="c2">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"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="jordan11.html#T9" title="JORDAN11:th.9">JORDAN11:9</a></span>;<br/>
<a NAME="E5:33_2"/><span class="lab"><font color="Green" title="E40">A63</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard7.html#T34" title="GOBOARD7:th.34">GOBOARD7:34</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A64</font></span>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="jordan9.html#T27" title="JORDAN9:th.27">JORDAN9:27</a></span>;<br/>
<a NAME="E7:33_2"/>
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></span>;<br/>
<a NAME="E8:33_2"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E5:33_2"><span class="lab"><font color="Green" title="E40">A63</font></span></a>, <a class="txt" href="#E2:33_2"><span class="lab"><font color="Green" title="E37">A60</font></span></a>, <a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="txt" href="#E4:33_2"><span class="lab"><font color="Green" title="E39">A62</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:33_2"/>
<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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_2"><span class="lab"><font color="Green" title="E41">A64</font></span></a>, <a class="ref" href="gobrd14.html#T37" title="GOBRD14:th.37">GOBRD14:37</a></span>; <a class="txt" 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>
<a NAME="E14:33"/>
 for <font color="Olive" title="b1">t</font> being    <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E13:33"><span class="lab"><font color="Green" title="E36">A59</font></span></a>, <a class="txt" href="#E12:33"><span class="lab"><font color="Green" title="E35">A10</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E15:33"/>
<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="c1">C</font>,<font color="Maroon" title="c2">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E30">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
