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

<span class="kw">set </span><font color="Maroon" title="c1">j</font> = 1;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">f</font> be   non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  iff <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <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> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">r</font> =  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>);<br/>
<a NAME="E1:27"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
 <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>) <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard5.html#D5" title="GOBOARD5:def.5">GOBOARD5:def 5</a></span>;<br/>
<a NAME="E2:27"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal8.html#T3" title="TOPREAL8:th.3">TOPREAL8:3</a></span>;<br/>
<a NAME="E3:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A2</font></span>: 
1 <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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">i2</font>, <font color="Maroon" title="c5">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:27"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:27"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/>
<a NAME="E7:27"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">i</font> =  <a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>;<br/>
<a NAME="E8:27"/><span class="lab"><font color="Green" title="E25">A6</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> <font color="Maroon" title="c2">f</font></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> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T42" title="SPRECT_2:th.42">SPRECT_2:42</a></span>;<br/>
<a NAME="E9:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</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> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T92" title="FINSEQ_6:th.92">FINSEQ_6:92</a></span>;<br/>
<a NAME="E10:27"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
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> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal8.html#T3" title="TOPREAL8:th.3">TOPREAL8:3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:27"/>
( <font color="Maroon" title="c2">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  implies <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <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> <font color="Maroon" title="c2">f</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <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> <font color="Maroon" title="c2">f</font></span>)</span> implies <font color="Maroon" title="c2">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c7">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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font>;<br/>
<a NAME="E1:27_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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_5.html#T14" title="SPRECT_5:th.14">SPRECT_5:14</a></span>;<br/>
<a NAME="E2:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A9</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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
 
<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="E3:27_1"/>
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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></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="E4:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A10</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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></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> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E5:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></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> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E6:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c2">f</font> <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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<a NAME="E7:27_1"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="sppol_2.html#T18" title="SPPOL_2:th.18">SPPOL_2:18</a></span>;<br/>
<a NAME="E8:27_1"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c2">f</font> <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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></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> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="finseq_5.html#T38" title="FINSEQ_5:th.38">FINSEQ_5:38</a></span>;<br/>
<a NAME="E9:27_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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E33">A14</font></span>: <font color="Maroon" title="c2">f</font> <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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="revrot_1.html#T10" title="REVROT_1:th.10">REVROT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</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"><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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
;<br/>
<a NAME="E11:27_1"/>
<font color="Maroon" title="c2">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard5.html#D5" title="GOBOARD5:def.5">GOBOARD5:def 5</a></span>;<br/>
<a NAME="E12:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="revrot_1.html#T28" title="REVROT_1:th.28">REVROT_1:28</a></span>;<br/>
<span class="kw">assume </span><a NAME="E13:27_1"/>
<font color="Maroon" title="c2">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <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> <font color="Maroon" title="c2">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">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="E15:27_1"/><span class="lab"><font color="Green" title="E35">A16</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="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:27_1"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:27_1"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
<font color="Maroon" title="c2">f</font> <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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font>)
 <span class="kw">and </span><br/><a NAME="E18:27_1"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<font color="Maroon" title="c2">f</font> <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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E8:27_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E12:27_1"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="jordan1i.html#T24" target="_self" title="JORDAN1I:th.24">Th26</a>, <a class="ref" href="finseq_4.html#T21" title="FINSEQ_4:th.21">FINSEQ_4:21</a></span>;<br/>
<a NAME="E19:27_1"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:27_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E20:27_1"/><span class="lab"><font color="Green" title="E40">A21</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">j</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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:27_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E21:27_1"/><span class="lab"><font color="Green" title="E41">A22</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">j</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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:27_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E22:27_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:27_1"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></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>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c9">j</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="#E18:27_1"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E20:27_1"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:27_1"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="txt" href="#E19:27_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="txt" href="#E21:27_1"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>
<br/>.= 
 <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> <font color="Maroon" title="c2">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E24:27_1"/>
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <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> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27_1"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E6:27_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="ref" href="sprect_2.html#T11" title="SPRECT_2:th.11">SPRECT_2:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E12:27"/><span class="lab"><font color="Green" title="E28">A23</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <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> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a> </span><br/>

<a NAME="E13:27"/><span class="lab"><font color="Green" title="E29">A24</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#D6" title="JORDAN5D:def.6">JORDAN5D:def 6</a></span>;<br/>
<a NAME="E14:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E15:27"/><span class="lab"><font color="Green" title="E31">A26</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="revrot_1.html#T33" title="REVROT_1:th.33">REVROT_1:33</a></span>;<br/>
<a NAME="E16:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A27</font></span>: 
<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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="jordan5d.html#D6" title="JORDAN5D:def.6">JORDAN5D:def 6</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font></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="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/>
<a NAME="E18:27"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="pscomp_1.html#T55" title="PSCOMP_1:th.55">PSCOMP_1:55</a></span>;<br/>
<a NAME="E19:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A28</font></span>: 
<font color="Maroon" title="c5">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="jordan1g.html#T6" title="JORDAN1G:th.6">JORDAN1G:6</a></span>;<br/>
<a NAME="E20:27"/><span class="lab"><font color="Green" title="E34">A29</font></span>: 
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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E21:27"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T90" title="FINSEQ_6:th.90">FINSEQ_6:90</a>, <a class="ref" href="sprect_2.html#T42" title="SPRECT_2:th.42">SPRECT_2:42</a></span>;<br/>
<a NAME="E22:27"/><span class="kw">then </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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a>, <a class="ref" href="sprect_2.html#T42" title="SPRECT_2:th.42">SPRECT_2:42</a></span>;<br/>
<a NAME="E23:27"/><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> 1</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"><span class="p3">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">i2</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="#E1:27"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E13:27"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E19:27"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/>
<a NAME="E24:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">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"><span class="p3">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">i2</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="ref" href="absvalue.html#T2" title="ABSVALUE:th.2">ABSVALUE:2</a></span>;<br/>
<a NAME="E25:27"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E12:27"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="pscomp_1.html#T55" title="PSCOMP_1:th.55">PSCOMP_1:55</a></span>;<br/>
<a NAME="E26:27"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="txt" href="#E20:27"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="txt" href="#E19:27"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a>, <a class="ref" href="xreal_1.html#T48" title="XREAL_1:th.48">XREAL_1:48</a></span>;<br/>
<a NAME="E27:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A31</font></span>: 
<span class="p1">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:27"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E28:27"/><span class="kw">then </span>
<font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
;<br/>
<a NAME="E29:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A32</font></span>: 
<font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E30:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A33</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E31:27"/><span class="lab"><font color="Green" title="E39">A34</font></span>: 
 <a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E32:27"/><span class="lab"><font color="Green" title="E40">A35</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal8.html#T3" title="TOPREAL8:th.3">TOPREAL8:3</a></span>;<br/>
<a NAME="E33:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A36</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>,1)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#T21" title="GOBOARD9:th.21">GOBOARD9:21</a></span>;<br/>
<a NAME="E34:27"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>,1)</span>)</span> <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="txt" href="#E32:27"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="ref" href="goboard9.html#T15" title="GOBOARD9:th.15">GOBOARD9:15</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E36:27"/><span class="lab"><font color="Green" title="E42">A37</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>,1)</span>)</span>
 <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/>
<a NAME="E37:27"/>
<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"><span class="p4">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p5">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<span class="p0">(<span class="default"><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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="txt" href="#E14:27"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E38:27"/><span class="kw">then </span>
( 1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>,1,<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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:27"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E19:27"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="txt" href="#E27:27"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E29:27"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E39:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A38</font></span>: 
 <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>,1) <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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:27"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="ref" href="jordan1h.html#T21" title="JORDAN1H:th.21">JORDAN1H:21</a></span>;<br/>
<a NAME="E40:27"/>
<span class="p1">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T146" title="XREAL_1:th.146">XREAL_1:146</a></span>;<br/>
<a NAME="E41:27"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:27"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="txt" href="#E27:27"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E29:27"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E42:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A39</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></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"><span class="p4">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p5">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<span class="p0">(<span class="default"><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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b2">s</font> <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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:27"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E39:27"><span class="lab"><font color="Green" title="E43">A38</font></span></a>, <a class="ref" href="goboard6.html#T24" title="GOBOARD6:th.24">GOBOARD6:24</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:27"><span class="lab"><font color="Green" title="E42">A37</font></span></a></span>;<br/>
<a NAME="E44:27"/><span class="lab"><font color="Green" title="E45">A40</font></span>: 
(  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp;  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a>, <a class="ref" href="jordan2c.html#T124" title="JORDAN2C:th.124">JORDAN2C:124</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">t</font>, <font color="Maroon" title="c10">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E46:27"/><span class="lab"><font color="Green" title="E46">A41</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c9">t</font>,<font color="Maroon" title="c10">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E47:27"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<span class="p5">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-max</a> <span class="p0">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">t</font>
 <span class="kw">and </span><br/><a NAME="E48:27"/>
<font color="Maroon" title="c9">t</font> <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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></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"><span class="p4">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p5">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<span class="p0">(<span class="default"><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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E49:27"/><span class="lab"><font color="Green" title="E47">A42</font></span>: 
<font color="Maroon" title="c10">s</font> <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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:27"><span class="lab"><font color="Green" title="E44">A39</font></span></a>, <a class="txt" href="#E36:27"><span class="lab"><font color="Green" title="E42">A37</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:27_3"/>
 <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c8">p</font> <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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c8">p</font></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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <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="c11">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:27_3"/><span class="lab"><font color="Green" title="E48">A43</font></span>: 
<font color="Maroon" title="c11">a</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> <font color="Maroon" title="c8">p</font></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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 <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/><a NAME="E5:27_3"/><span class="lab"><font color="Green" title="E49">A44</font></span>: 
<font color="Maroon" title="c11">a</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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_3"><span class="lab"><font color="Green" title="E48">A43</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="E6:27_3"/><span class="lab"><font color="Green" title="E50">A45</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c8">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_3"><span class="lab"><font color="Green" title="E48">A43</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c11">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_3"><span class="lab"><font color="Green" title="E48">A43</font></span></a></span>;<br/><a NAME="E8:27_3"/>
<font color="Maroon" title="c12">a</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> <font color="Maroon" title="c8">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_3"><span class="lab"><font color="Green" title="E50">A45</font></span></a>, <a class="ref" href="topreal1.html#D12" title="TOPREAL1:def.12">TOPREAL1:def 12</a></span>;<br/><a NAME="E9:27_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">a</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> <font color="Maroon" title="c10">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:27"><span class="lab"><font color="Green" title="E46">A41</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/><a NAME="E10:27_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A46</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:27"><span class="lab"><font color="Green" title="E47">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:27_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan5d.html#K6" title="JORDAN5D:func.6">i_e_s</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>,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="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="txt" href="#E31:27"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="txt" href="#E20:27"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><a NAME="E12:27_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">a</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:27"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="txt" href="#E9:27"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E10:27_3"><span class="lab"><font color="Green" title="E51">A46</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/><span class="kw">hence </span><a NAME="E13:27_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_3"><span class="lab"><font color="Green" title="E49">A44</font></span></a>, <a class="ref" href="pscomp_1.html#T24" title="PSCOMP_1:th.24">PSCOMP_1:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E51:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A47</font></span>: 
 <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c8">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T128" title="JORDAN2C:th.128">JORDAN2C:128</a></span>;<br/>
<a NAME="E52:27"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K6" title="TOPREAL1:func.6">south_halfline</a> <font color="Maroon" title="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#T38" title="TOPREAL1:th.38">TOPREAL1:38</a></span>;<br/>
<a NAME="E53:27"/><span class="kw">then </span>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</font>,<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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:27"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="txt" href="#E36:27"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="txt" href="#E51:27"><span class="lab"><font color="Green" title="E48">A47</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E54:27"/><span class="kw">then </span>
 <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c2">f</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> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>) is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:27"><span class="lab"><font color="Green" title="E45">A40</font></span></a>, <a class="ref" href="goboard9.html#T1" title="GOBOARD9:th.1">GOBOARD9:1</a>, <a class="ref" href="jordan1h.html#T41" title="JORDAN1H:th.41">JORDAN1H:41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E55:27"/>
<font color="Maroon" title="c2">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T40" title="JORDAN1H:th.40">JORDAN1H:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
