<?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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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> holds   <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p2">(<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>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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></span>)</span></span>)</span></span>)</span>,1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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="Red">::  thesis: </font></span></a><span class="hide">  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</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 class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span></span>)</span>,1</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">f</font> =  <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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E2:10"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 <span class="kw">and </span><br/><a NAME="E3:10"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c5">j</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">and </span><br/><a NAME="E4:10"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
 <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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">(<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">j</font>,1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1d.html#T32" title="JORDAN1D:th.32">JORDAN1D:32</a></span>;<br/>
<a NAME="E5:10"/><span class="lab"><font color="Green" title="E8">A4</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">set </span><font color="Maroon" title="c6">k</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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>;<br/>
<a NAME="E6:10"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
 <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</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="ref" href="sprect_2.html#T46" title="SPRECT_2:th.46">SPRECT_2:46</a></span>;<br/>
<a NAME="E7:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A6</font></span>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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; <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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>  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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="ref" href="finseq_4.html#T29" title="FINSEQ_4:th.29">FINSEQ_4:29</a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a></span>;<br/>
<a NAME="E8:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A7</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"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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>  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E12">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:10_1"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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="ref" href="jordan1i.html#T5" target="_self" title="JORDAN1I:th.5">Th5</a></span>;<br/><a NAME="E2:10_1"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="#E6:10"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="finseq_3.html#T33" title="FINSEQ_3:th.33">FINSEQ_3:33</a></span>;<br/><span class="kw">assume </span><a NAME="E3:10_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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> <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="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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:10_1"/><span class="kw">then </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> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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="#E8:10"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></span>;<br/><a NAME="E5:10_1"/><span class="kw">then </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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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="#E2:10_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E6:10_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="finseq_4.html#T34" title="FINSEQ_4:th.34">FINSEQ_4:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:10"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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> <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">by</span> <span class="lab"><a class="txt" href="#E6:10"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></span>;<br/>
<a NAME="E11:10"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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> <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="#E9:10"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E12:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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="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">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="E13:10"/><span class="lab"><font color="Green" title="E14">A12</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"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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">(<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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <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</span>)</span>,1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E8:10"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/>
<a NAME="E14:10"/>
<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"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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">(<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">j</font>,1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E15:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A13</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"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</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="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">j</font>,1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T52" title="JORDAN1H:th.52">JORDAN1H:52</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">p</font> =  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <font color="Maroon" title="c2">C</font>;<br/>
<a NAME="E16:10"/><span class="lab"><font color="Green" title="E16">A14</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="E17:10"/><span class="lab"><font color="Green" title="E17">A15</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#T13" title="JORDAN8:th.13">JORDAN8:13</a></span>;<br/>
<a NAME="E18:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A16</font></span>: 
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="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E19:10"/><span class="lab"><font color="Green" title="E19">A17</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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="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="E20:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A18</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="#E12:10"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E21:10"/><span class="lab"><font color="Green" title="E21">A19</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="#E12:10"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="txt" href="#E19:10"><span class="lab"><font color="Green" title="E19">A17</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">kj</font>, <font color="Maroon" title="c9">ki</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E23:10"/><span class="lab"><font color="Green" title="E22">A20</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">kj</font>,<font color="Maroon" title="c9">ki</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="E24:10"/><span class="lab"><font color="Green" title="E23">A21</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p5">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<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="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="c8">kj</font>,<font color="Maroon" title="c9">ki</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/>
<a NAME="E25:10"/><span class="lab"><font color="Green" title="E24">A22</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">kj</font>,<font color="Maroon" title="c9">ki</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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</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> &amp; <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p5">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<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="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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</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="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">kj</font>,<font color="Maroon" title="c9">ki</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:10"><span class="lab"><font color="Green" title="E22">A20</font></span></a>, <a class="txt" href="#E24:10"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="ref" href="jordan1h.html#T52" title="JORDAN1H:th.52">JORDAN1H:52</a></span>;<br/>
<a NAME="E26:10"/><span class="lab"><font color="Green" title="E25">A23</font></span>: 
<font color="Maroon" title="c9">ki</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="#E23:10"><span class="lab"><font color="Green" title="E22">A20</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E27:10"/><span class="lab"><font color="Green" title="E26">A24</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">kj</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:10"><span class="lab"><font color="Green" title="E22">A20</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E28:10"/>
 <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="E29:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A25</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">j</font>,1</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">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E18:10"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>;<br/>
<a NAME="E30:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A26</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <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</span>)</span>,1</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">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/>
<a NAME="E31:10"/><span class="lab"><font color="Green" title="E29">A27</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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="ref" href="jordan1i.html#T5" target="_self" title="JORDAN1I:th.5">Th5</a></span>;<br/>
<a NAME="E32:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A28</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p0">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p4">(<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="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>  <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="#E8:10"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E12:10"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="jordan1e.html#T25" title="JORDAN1E:th.25">JORDAN1E:25</a></span>;<br/>
<a NAME="E33:10"/><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">j</font>,1</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="c8">kj</font>,<font color="Maroon" title="c9">ki</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="#E4:10"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E24:10"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/>
<a NAME="E34:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A29</font></span>: 
<font color="Maroon" title="c9">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:10"><span class="lab"><font color="Green" title="E22">A20</font></span></a>, <a class="txt" href="#E29:10"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></span>;<br/>
<a NAME="E35:10"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">j</font>,1</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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</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="#E29:10"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="jordan1h.html#T52" title="JORDAN1H:th.52">JORDAN1H:52</a></span>;<br/>
<a NAME="E36:10"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">ki</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">kj</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E20:10"><span class="lab"><font color="Green" title="E20">A18</font></span></a>, <a class="txt" href="#E15:10"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="txt" href="#E25:10"><span class="lab"><font color="Green" title="E24">A22</font></span></a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a>, <a class="ref" href="goboard5.html#T13" title="GOBOARD5:th.13">GOBOARD5:13</a></span>;<br/>
<a NAME="E37:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A30</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">kj</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:10"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>;<br/>
<a NAME="E38:10"/><span class="lab"><font color="Green" title="E33">A31</font></span>: 
<font color="Maroon" title="c8">kj</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="#E23:10"><span class="lab"><font color="Green" title="E22">A20</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E39:10"/>
2 <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">by</span> <span class="lab"><a class="ref" href="goboard7.html#T36" title="GOBOARD7:th.36">GOBOARD7:36</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E40:10"/><span class="kw">then </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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p5">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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> <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="#E32:10"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="txt" href="#E21:10"><span class="lab"><font color="Green" title="E21">A19</font></span></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a>, <a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></span>;<br/>
<a NAME="E41:10"/><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">j</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"><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="c8">kj</font>,<font color="Maroon" title="c9">ki</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="#E4:10"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E24:10"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="ref" href="pscomp_1.html#T118" title="PSCOMP_1:th.118">PSCOMP_1:118</a></span>;<br/>
<a NAME="E42:10"/><span class="kw">then </span>
<font color="Maroon" title="c8">kj</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E34:10"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E26:10"><span class="lab"><font color="Green" title="E25">A23</font></span></a>, <a class="txt" href="#E38:10"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>;<br/>
<a NAME="E43:10"/><span class="kw">then </span>
<font color="Maroon" title="c8">kj</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="c5">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:10"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="ref" href="seqm_3.html#T81" title="SEQM_3:th.81">SEQM_3:81</a></span>;<br/>
<a NAME="E44:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A32</font></span>: 
<font color="Maroon" title="c8">kj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
;<br/>
<a NAME="E45:10"/><span class="kw">then </span>
<font color="Maroon" title="c8">kj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:10"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="ref" href="nat_d.html#T39" title="NAT_D:th.39">NAT_D:39</a></span>;<br/>
<a NAME="E46:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A33</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1</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">by</span> <span class="lab"><a class="txt" href="#E18:10"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="txt" href="#E27:10"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E38:10"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>;<br/>
<a NAME="E47:10"/>
<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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p5">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<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="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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:10"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="txt" href="#E34:10"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E27:10"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E44:10"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="ref" href="nat_d.html#T39" title="NAT_D:th.39">NAT_D:39</a></span>;<br/>
<a NAME="E48:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A34</font></span>: 
 <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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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="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="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="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E31:10"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="txt" href="#E12:10"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="txt" href="#E46:10"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="txt" href="#E30:10"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="txt" href="#E13:10"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E37">A35</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:10_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:10"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E2:10_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A36</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<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/><span class="kw">assume </span><a NAME="E3:10_2"/><span class="lab"><font color="Green" title="E39">A37</font></span>: 
not  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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="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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:10_2"/><span class="lab"><font color="Green" title="E40">A38</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:10"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E44:10"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="ref" href="nat_d.html#T39" title="NAT_D:th.39">NAT_D:39</a></span>;<br/><a NAME="E5:10_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T51" title="NAT_D:th.51">NAT_D:51</a></span>;<br/><a NAME="E6:10_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:10_2"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="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="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <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</span>)</span>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="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="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_2"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="txt" href="#E4:10_2"><span class="lab"><font color="Green" title="E40">A38</font></span></a>, <a class="ref" href="goboard5.html#T22" title="GOBOARD5:th.22">GOBOARD5:22</a></span>;<br/><a NAME="E8:10_2"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="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="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">j</font>,2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E9:10_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A39</font></span>: 
not  <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="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="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">j</font>,2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:10"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="txt" href="#E3:10_2"><span class="lab"><font color="Green" title="E39">A37</font></span></a></span>;<br/><a NAME="E10:10_2"/>
 <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="E11:10_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A40</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:10"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:10_2"/><span class="lab"><font color="Green" title="E43">A41</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="E13:10_2"/><span class="lab"><font color="Green" title="E44">A42</font></span>: 
<font color="Maroon" title="c5">j</font> <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="txt" href="#E27:10"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E38:10"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E44:10"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="ref" href="nat_d.html#T39" title="NAT_D:th.39">NAT_D:39</a></span>;<br/><a NAME="E14:10_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A43</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,2</span>)</span> <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> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_2"><span class="lab"><font color="Green" title="E40">A38</font></span></a>, <a class="ref" href="jordan8.html#T16" title="JORDAN8:th.16">JORDAN8:16</a></span>;<br/><a NAME="E15:10_2"/>
( <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">j</font>,2</span>)</span> <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> <font color="Maroon" title="c2">C</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <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> <font color="Maroon" title="c2">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a>, <a class="ref" href="jordan8.html#T16" title="JORDAN8:th.16">JORDAN8:16</a></span>;<br/><a NAME="E16:10_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A44</font></span>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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#NR4" title="XXREAL_0:NR.4">&gt;</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">j</font>,2</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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#NR3" title="XXREAL_0:NR.3">&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="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,2</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="#E9:10_2"><span class="lab"><font color="Green" title="E41">A39</font></span></a>, <a class="txt" href="#E14:10_2"><span class="lab"><font color="Green" title="E45">A43</font></span></a>, <a class="ref" href="goboard7.html#T9" title="GOBOARD7:th.9">GOBOARD7:9</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:10_2_1"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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#NR3" title="XXREAL_0:NR.3">&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="c5">j</font> <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>  or <span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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#NR4" title="XXREAL_0:NR.4">&gt;</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">j</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E4:10_2"><span class="lab"><font color="Green" title="E40">A38</font></span></a>, <a class="txt" href="#E13:10_2"><span class="lab"><font color="Green" title="E44">A42</font></span></a>, <a class="txt" href="#E16:10_2"><span class="lab"><font color="Green" title="E46">A44</font></span></a>, <a class="txt" href="#E11:10_2"><span class="lab"><font color="Green" title="E42">A40</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:10_2_1_1"/><span class="lab"><font color="Green" title="E47">A45</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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#NR3" title="XXREAL_0:NR.3">&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="c5">j</font> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:10_2_1_1"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:10"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="txt" href="#E12:10"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="txt" href="#E48:10"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="ref" href="jordan9.html#T33" title="JORDAN9:th.33">JORDAN9:33</a></span>;<br/><a NAME="E3:10_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:10_2_1_1"/><span class="lab"><font color="Green" title="E48">A46</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c10">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">by</span> <span class="lab"><a class="txt" href="#E5:10_2_1_1"><span class="lab"><font color="Green" title="E48">A46</font></span></a></span>;<br/><a NAME="E7:10_2_1_1"/><span class="lab"><font color="Green" title="E49">A47</font></span>: 
<font color="Maroon" title="c11">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="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_2_1_1"><span class="lab"><font color="Green" title="E48">A46</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:10_2_1_1"/><span class="lab"><font color="Green" title="E50">A48</font></span>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_2_1_1"><span class="lab"><font color="Green" title="E48">A46</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E9:10_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A49</font></span>: 
<font color="Maroon" title="c11">c</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/><a NAME="E10:10_2_1_1"/><span class="lab"><font color="Green" title="E52">A50</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <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="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> &amp; 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="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="#E2:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E17:10"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="txt" href="#E12:10_2"><span class="lab"><font color="Green" title="E43">A41</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:10_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">c</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="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="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default">1 <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="#E4:10_2"><span class="lab"><font color="Green" title="E40">A38</font></span></a>, <a class="txt" href="#E7:10_2_1_1"><span class="lab"><font color="Green" title="E49">A47</font></span></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a></span>;<br/><a NAME="E12:10_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">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="txt" href="#E14:10_2"><span class="lab"><font color="Green" title="E45">A43</font></span></a>, <a class="txt" href="#E8:10_2_1_1"><span class="lab"><font color="Green" title="E50">A48</font></span></a>, <a class="txt" href="#E9:10_2_1_1"><span class="lab"><font color="Green" title="E51">A49</font></span></a>, <a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E13:10_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A51</font></span>: 
<font color="Maroon" title="c11">c</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"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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="ref" href="pscomp_1.html#T118" title="PSCOMP_1:th.118">PSCOMP_1:118</a></span>;<br/><a NAME="E14:10_2_1_1"/>
<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="c5">j</font> <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="Maroon" title="c11">c</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_2"><span class="lab"><font color="Green" title="E40">A38</font></span></a>, <a class="txt" href="#E7:10_2_1_1"><span class="lab"><font color="Green" title="E49">A47</font></span></a>, <a class="txt" href="#E10:10_2_1_1"><span class="lab"><font color="Green" title="E52">A50</font></span></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a></span>;<br/><span class="kw">hence </span><a NAME="E15:10_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_1_1"><span class="lab"><font color="Green" title="E47">A45</font></span></a>, <a class="txt" href="#E13:10_2_1_1"><span class="lab"><font color="Green" title="E53">A51</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:10_2_1_2"/><span class="lab"><font color="Green" title="E47">A52</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</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#NR4" title="XXREAL_0:NR.4">&gt;</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">j</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:10_2_1_2"/>
 <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</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="ref" href="jordan1a.html#T74" title="JORDAN1A:th.74">JORDAN1A:74</a>, <a class="ref" href="sprect_1.html#T14" title="SPRECT_1:th.14">SPRECT_1:14</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:10_2_1_2"/><span class="lab"><font color="Green" title="E48">A53</font></span>: 
<font color="Maroon" title="c10">r</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#K24" title="PSCOMP_1:func.24">S-max</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:10_2_1_2"/><span class="lab"><font color="Green" title="E49">A54</font></span>: 
<font color="Maroon" title="c10">r</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="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c10">r</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">by</span> <span class="lab"><a class="txt" href="#E4:10_2_1_2"><span class="lab"><font color="Green" title="E48">A53</font></span></a></span>;<br/><a NAME="E7:10_2_1_2"/>
<font color="Maroon" title="c11">r</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#K24" title="PSCOMP_1:func.24">S-max</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="#E4:10_2_1_2"><span class="lab"><font color="Green" title="E48">A53</font></span></a>, <a class="txt" href="#E5:10_2_1_2"><span class="lab"><font color="Green" title="E49">A54</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:10_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">r</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="ref" href="jordan1a.html#T105" title="JORDAN1A:th.105">JORDAN1A:105</a>, <a class="ref" href="pscomp_1.html#T121" title="PSCOMP_1:th.121">PSCOMP_1:121</a></span>;<br/><a NAME="E9:10_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">r</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> <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="#E5:10_2_1_2"><span class="lab"><font color="Green" title="E49">A54</font></span></a>, <a class="ref" href="sprect_2.html#T15" title="SPRECT_2:th.15">SPRECT_2:15</a></span>;<br/><a NAME="E10:10_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="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>)</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> <font color="Maroon" title="c11">r</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T118" title="PSCOMP_1:th.118">PSCOMP_1:118</a></span>;<br/><span class="kw">hence </span><a NAME="E11:10_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E1:10_2_1_2"><span class="lab"><font color="Green" title="E47">A52</font></span></a>, <a class="txt" href="#E4:10_2_1_2"><span class="lab"><font color="Green" title="E48">A53</font></span></a>, <a class="ref" href="topreal1.html#D14" title="TOPREAL1:def.14">TOPREAL1:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E50:10"/>
 <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="jordan1h.html#T52" title="JORDAN1H:th.52">JORDAN1H:52</a></span>;<br/>
<a NAME="E51:10"/><span class="kw">then </span>
 <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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="#E31:10"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="txt" href="#E12:10"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="txt" href="#E49:10"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="ref" href="jordan1h.html#T29" title="JORDAN1H:th.29">JORDAN1H:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E52:10"/>
 <a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</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 class="p2">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<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>)</span></span>)</span>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="jordan1i.html#T7" target="_self" title="JORDAN1I:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
