<?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#K25" title="PSCOMP_1:func.25">S-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#K25" title="PSCOMP_1:func.25">S-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="E2:28"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-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#T88" title="JORDAN1A:th.88">JORDAN1A:88</a>, <a class="ref" href="pscomp_1.html#T58" title="PSCOMP_1:th.58">PSCOMP_1:58</a></span>;<br/>
<a NAME="E3:28"/><span class="lab"><font color="Green" title="E26">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#K6" title="TOPREAL1:func.6">south_halfline</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-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="#E2:28"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">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#K6" title="TOPREAL1:func.6">south_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-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="E5:28"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-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#K25" title="PSCOMP_1:func.25">S-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#K25" title="PSCOMP_1:func.25">S-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="E6:28"/><span class="lab"><font color="Green" title="E29">A5</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="E7:28"/><span class="lab"><font color="Green" title="E30">A6</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="E8:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A7</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="E9:28"/>
<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="#E3:28"><span class="lab"><font color="Green" title="E26">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="E11:28"/><span class="lab"><font color="Green" title="E32">A8</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="E12:28"/><span class="lab"><font color="Green" title="E33">A9</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="E13:28"/><span class="lab"><font color="Green" title="E34">A10</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#K25" title="PSCOMP_1:func.25">S-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="E14:28"/><span class="lab"><font color="Green" title="E35">A11</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="#E11:28"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E36">A12</font></span>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-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> (1,2)</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="#E8:28"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="jordan8.html#T13" title="JORDAN8:th.13">JORDAN8:13</a></span>
;<br/>
<a NAME="E16:28"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-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#K17" title="PSCOMP_1:func.17">S-most</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T58" title="PSCOMP_1:th.58">PSCOMP_1:58</a></span>;<br/>
<span class="kw">thus </span><a NAME="E17:28"/><span class="lab"><font color="Green" title="E38">A14</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="#E11:28"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E33">A9</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#K25" title="PSCOMP_1:func.25">S-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="E18:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A15</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="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E4:28"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E13:28"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E16:28"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="jordan1a.html#T80" title="JORDAN1A:th.80">JORDAN1A:80</a>, <a class="ref" href="sppol_1.html#T40" title="SPPOL_1:th.40">SPPOL_1:40</a></span>;<br/>
<a NAME="E19:28"/><span class="lab"><font color="Green" title="E40">A16</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="E21:28"/><span class="lab"><font color="Green" title="E41">A17</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="E22:28"/><span class="lab"><font color="Green" title="E42">A18</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="E23:28"/><span class="lab"><font color="Green" title="E43">A19</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="E24:28"/><span class="lab"><font color="Green" title="E44">A20</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="E25:28"/><span class="lab"><font color="Green" title="E45">A21</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="#E11:28"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/>
<a NAME="E26:28"/><span class="lab"><font color="Green" title="E46">A22</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="#E21:28"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E27:28"/><span class="lab"><font color="Green" title="E47">A23</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="#E23:28"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E28:28"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
<font color="Maroon" title="c7">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i2</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="E29:28"/><span class="lab"><font color="Green" title="E49">A25</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="#E23:28"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E30:28"/><span class="lab"><font color="Green" title="E50">A26</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="#E21:28"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E31:28"/><span class="lab"><font color="Green" title="E51">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="#E21:28"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E32:28"/>
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-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="#E3:28"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="jordan1a.html#T84" title="JORDAN1A:th.84">JORDAN1A:84</a>, <a class="ref" href="pscomp_1.html#T58" title="PSCOMP_1:th.58">PSCOMP_1:58</a></span>;<br/>
<a NAME="E33:28"/><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#K18" title="EUCLID:func.18">`2</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> (<font color="Maroon" title="c5">i1</font>,1)</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="#E22:28"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E26:28"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E30:28"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="jordan1a.html#T72" title="JORDAN1A:th.72">JORDAN1A:72</a></span>;<br/>
<a NAME="E34:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A28</font></span>: 
1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:28"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E30:28"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E31:28"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>;<br/>
<a NAME="E35:28"/><span class="lab"><font color="Green" title="E53">A29</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="#E21:28"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E36:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A30</font></span>: 
<font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:28"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E37:28"/><span class="lab"><font color="Green" title="E55">A31</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#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E4:28"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E13:28"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E17:28"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E16:28"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="jordan1a.html#T80" title="JORDAN1A:th.80">JORDAN1A:80</a>, <a class="ref" href="sppol_1.html#T40" title="SPPOL_1:th.40">SPPOL_1:40</a></span>;<br/>
<a NAME="E38:28"/><span class="lab"><font color="Green" title="E56">A32</font></span>: 
<font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:28_2"/>
<font color="Maroon" title="c6">j1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">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/>

<a NAME="E2:28_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j2</font> or <font color="Maroon" title="c8">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j1</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="E3:28_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:28"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E37:28"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E26:28"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E30:28"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E35:28"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E29:28"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E27:28"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E31:28"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</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="E39:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A33</font></span>: 
<font color="Maroon" title="c7">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E21:28"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E22:28"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E30:28"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E34:28"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="jordan10.html#T3" title="JORDAN10:th.3">JORDAN10:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E40:28"/>
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="#E23:28"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E41:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A34</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="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#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E21:28"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E22:28"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E30:28"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E35:28"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E29:28"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E27:28"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E31:28"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E34:28"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="txt" href="#E28:28"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="jordan10.html#T3" title="JORDAN10:th.3">JORDAN10:3</a>, <a class="ref" href="jordan1a.html#T18" title="JORDAN1A:th.18">JORDAN1A:18</a></span>;<br/>
<a NAME="E42:28"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:28"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="topreal1.html#T3" title="TOPREAL1:th.3">TOPREAL1:3</a></span>;<br/>
<a NAME="E43:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A35</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <font color="Maroon" title="c2">C</font></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> <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"><font color="Maroon" title="c7">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="#E4:28"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E11:28"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E21:28"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E22:28"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E38:28"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E36:28"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="jordan10.html#T3" title="JORDAN10:th.3">JORDAN10:3</a>, <a class="ref" href="topreal1.html#D12" title="TOPREAL1:def.12">TOPREAL1:def 12</a></span>;<br/>
<a NAME="E44:28"/>
<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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="#E13:28"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E41:28"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="ref" href="topreal1.html#T3" title="TOPREAL1:th.3">TOPREAL1:3</a></span>;<br/>
<a NAME="E45:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A36</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> (<font color="Maroon" title="c7">i2</font>,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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-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="#E4:28"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E38:28"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E36:28"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="topreal1.html#D12" title="TOPREAL1:def.12">TOPREAL1:def 12</a></span>;<br/>
<a NAME="E46:28"/><span class="lab"><font color="Green" title="E61">A37</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="#E23:28"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E47:28"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="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="#E7:28"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E48:28"/><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> (1,<font color="Maroon" title="c6">j1</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#K25" title="PSCOMP_1:func.25">S-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="#E6:28"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E36:28"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E15:28"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="sprect_3.html#T12" title="SPRECT_3:th.12">SPRECT_3:12</a></span>;<br/>
<a NAME="E49:28"/><span class="kw">then </span>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-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> (<font color="Maroon" title="c7">i2</font>,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"><font color="Maroon" title="c7">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c6">j1</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="c6">j1</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="#E36:28"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E15:28"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E45:28"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E43:28"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E28">A4</font></span></a></span>;<br/>
<a NAME="E50:28"/><span class="kw">then </span>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-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>,<font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c6">j1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E36:28"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E46:28"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="txt" href="#E39:28"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E31">A7</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="E51:28"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-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="#E11:28"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E19:28"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E21:28"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E22:28"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E38:28"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E34:28"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a>, <a class="ref" href="jordan10.html#T3" title="JORDAN10:th.3">JORDAN10:3</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/>


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