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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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">  for <font color="Olive" title="b1">C</font> being   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>   non  <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a>   non  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  ex <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp;  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Olive" title="b2">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</font> be   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>   non  <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a>   non  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp;  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>) )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:24"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1a.html#T87" title="JORDAN1A:th.87">JORDAN1A:87</a>, <a class="ref" href="pscomp_1.html#T50" title="PSCOMP_1:th.50">PSCOMP_1:50</a></span>;<br/>
<a NAME="E3:24"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:24"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <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/>
<a NAME="E6:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E7:24"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/>
<a NAME="E8:24"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
 <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>;<br/>
<a NAME="E9:24"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#T10" title="JORDAN8:th.10">JORDAN8:10</a></span>;<br/>
<a NAME="E10:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A8</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:24"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <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/>
<span class="lab"><font color="Green" title="E32">A10</font></span>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c2">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="jordan8.html#T12" title="JORDAN8:th.12">JORDAN8:12</a></span>
;<br/>
<a NAME="E13:24"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></span>;<br/>
<a NAME="E14:24"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
 <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">E-most</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T50" title="PSCOMP_1:th.50">PSCOMP_1:50</a></span>;<br/>
<a NAME="E15:24"/>
<font color="Maroon" title="c3">p</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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E17:24"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E18:24"/><span class="lab"><font color="Green" title="E36">A14</font></span>: 
<font color="Maroon" title="c4">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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E19:24"/><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<font color="Maroon" title="c3">p</font> <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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">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/>
<span class="kw">take </span>
<font color="Maroon" title="c4">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp;  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>) )</span><br/>

<a NAME="E20:24"/><span class="lab"><font color="Green" title="E38">A16</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">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="#E17:24"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E21:24"/><span class="lab"><font color="Green" title="E39">A17</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span><br/>

<a NAME="E22:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A18</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E19:24"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E14:24"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E20:24"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="ref" href="jordan1a.html#T79" title="JORDAN1A:th.79">JORDAN1A:79</a>, <a class="ref" href="sppol_1.html#T41" title="SPPOL_1:th.41">SPPOL_1:41</a></span>;<br/>
<a NAME="E23:24"/><span class="lab"><font color="Green" title="E41">A19</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E19:24"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E21:24"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="txt" href="#E14:24"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E20:24"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="ref" href="jordan1a.html#T79" title="JORDAN1A:th.79">JORDAN1A:79</a>, <a class="ref" href="sppol_1.html#T41" title="SPPOL_1:th.41">SPPOL_1:41</a></span>;<br/>
<a NAME="E24:24"/><span class="lab"><font color="Green" title="E42">A20</font></span>: 
 <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">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="c2">C</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan9.html#D1" title="JORDAN9:def.1">JORDAN9:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">i1</font>, <font color="Maroon" title="c6">j1</font>, <font color="Maroon" title="c7">i2</font>, <font color="Maroon" title="c8">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E26:24"/><span class="lab"><font color="Green" title="E43">A21</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E27:24"/><span class="lab"><font color="Green" title="E44">A22</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">j1</font>)
 <span class="kw">and </span><br/><a NAME="E28:24"/><span class="lab"><font color="Green" title="E45">A23</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c8">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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E29:24"/><span class="lab"><font color="Green" title="E46">A24</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c8">j2</font>)
 <span class="kw">and </span><br/><a NAME="E30:24"/><span class="lab"><font color="Green" title="E47">A25</font></span>: 
( ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">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="c8">j2</font> ) or ( <font color="Maroon" title="c5">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="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> ) or ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> ) or ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="#E17:24"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/>
<a NAME="E31:24"/><span class="lab"><font color="Green" title="E48">A26</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E32:24"/><span class="lab"><font color="Green" title="E49">A27</font></span>: 
<font color="Maroon" title="c6">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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E33:24"/><span class="lab"><font color="Green" title="E50">A28</font></span>: 
<font color="Maroon" title="c8">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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:24"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E34:24"/><span class="lab"><font color="Green" title="E51">A29</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:24"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E35:24"/><span class="lab"><font color="Green" title="E52">A30</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E36:24"/>
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="jordan1a.html#T83" title="JORDAN1A:th.83">JORDAN1A:83</a>, <a class="ref" href="pscomp_1.html#T50" title="PSCOMP_1:th.50">PSCOMP_1:50</a></span>;<br/>
<a NAME="E37:24"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">j1</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c6">j1</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:24"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E22:24"><span class="lab"><font color="Green" title="E40">A18</font></span></a>, <a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E35:24"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E32:24"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="jordan1a.html#T71" title="JORDAN1A:th.71">JORDAN1A:71</a></span>;<br/>
<a NAME="E38:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A31</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:24"><span class="lab"><font color="Green" title="E48">A26</font></span></a>, <a class="txt" href="#E35:24"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E32:24"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>;<br/>
<a NAME="E39:24"/><span class="lab"><font color="Green" title="E54">A32</font></span>: 
<font color="Maroon" title="c5">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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E40:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A33</font></span>: 
<font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:24"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E41:24"/><span class="lab"><font color="Green" title="E56">A34</font></span>: 
<font color="Maroon" title="c7">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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:24"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E42:24"/><span class="lab"><font color="Green" title="E57">A35</font></span>: 
<font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_2"/>
<font color="Maroon" title="c5">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i2</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/>

<a NAME="E2:24_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">i2</font> or <font color="Maroon" title="c7">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:24"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E30:24"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="txt" href="#E22:24"><span class="lab"><font color="Green" title="E40">A18</font></span></a>, <a class="txt" href="#E23:24"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="txt" href="#E31:24"><span class="lab"><font color="Green" title="E48">A26</font></span></a>, <a class="txt" href="#E39:24"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E34:24"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="txt" href="#E41:24"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E35:24"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E33:24"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</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="E43:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A36</font></span>: 
<font color="Maroon" title="c8">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="txt" href="#E27:24"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E28:24"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E30:24"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="txt" href="#E32:24"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E38:24"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="jordan10.html#T1" title="JORDAN10:th.1">JORDAN10:1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E44:24"/><span class="lab"><font color="Green" title="E59">A37</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:24"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E45:24"/>
<font color="Maroon" title="c8">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">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="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E46:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A38</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="txt" href="#E27:24"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E28:24"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E30:24"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="txt" href="#E31:24"><span class="lab"><font color="Green" title="E48">A26</font></span></a>, <a class="txt" href="#E39:24"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E44:24"><span class="lab"><font color="Green" title="E59">A37</font></span></a>, <a class="txt" href="#E32:24"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E42:24"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E38:24"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="jordan10.html#T1" title="JORDAN10:th.1">JORDAN10:1</a>, <a class="ref" href="jordan1a.html#T19" title="JORDAN1A:th.19">JORDAN1A:19</a></span>;<br/>
<a NAME="E47:24"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:24"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E20:24"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/>
<a NAME="E48:24"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E17:24"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="txt" href="#E27:24"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E28:24"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E30:24"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="txt" href="#E42:24"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E40:24"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="jordan10.html#T1" title="JORDAN10:th.1">JORDAN10:1</a>, <a class="ref" href="topreal1.html#D11" title="TOPREAL1:def.11">TOPREAL1:def 11</a></span>;<br/>
<a NAME="E49:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A39</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="txt" href="#E27:24"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E28:24"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E30:24"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="txt" href="#E34:24"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="txt" href="#E35:24"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E32:24"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E42:24"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E40:24"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a>, <a class="ref" href="jordan10.html#T1" title="JORDAN10:th.1">JORDAN10:1</a></span>;<br/>
<a NAME="E50:24"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:24"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E20:24"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="txt" href="#E46:24"><span class="lab"><font color="Green" title="E60">A38</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/>
<a NAME="E51:24"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c8">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E42:24"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E40:24"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="topreal1.html#D11" title="TOPREAL1:def.11">TOPREAL1:def 11</a></span>;<br/>
<a NAME="E52:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A40</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c8">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:24"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="txt" href="#E33:24"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="txt" href="#E44:24"><span class="lab"><font color="Green" title="E59">A37</font></span></a>, <a class="txt" href="#E42:24"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E40:24"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/>
<a NAME="E53:24"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <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="txt" href="#E9:24"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E54:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A41</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<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="E55:24"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<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/>
<a NAME="E56:24"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">i1</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E40:24"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="sprect_3.html#T13" title="SPRECT_3:th.13">SPRECT_3:13</a></span>;<br/>
<a NAME="E57:24"/><span class="kw">then </span>
 <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</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"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c8">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</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"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:24"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E52:24"><span class="lab"><font color="Green" title="E62">A40</font></span></a>, <a class="txt" href="#E49:24"><span class="lab"><font color="Green" title="E61">A39</font></span></a>, <a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E28">A6</font></span></a></span>;<br/>
<a NAME="E58:24"/><span class="kw">then </span>
 <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c8">j2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:24"><span class="lab"><font color="Green" title="E59">A37</font></span></a>, <a class="txt" href="#E42:24"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E40:24"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E43:24"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="txt" href="#E54:24"><span class="lab"><font color="Green" title="E63">A41</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="gobrd11.html#T32" title="GOBRD11:th.32">GOBRD11:32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E59:24"/>
 <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E24:24"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E26:24"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="txt" href="#E27:24"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E28:24"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E30:24"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="txt" href="#E42:24"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E38:24"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a>, <a class="ref" href="jordan10.html#T1" title="JORDAN10:th.1">JORDAN10:1</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>
