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

<b>let </b><font color="Maroon">f</font> be  <a href="funct_1.html#V3">V7</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1"/>
(  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 or  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>suppose </b></a><a NAME="E1:50_1_1"/>
 <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:50_1_1"/>
( <font color="Maroon">f</font> is <a href="sprect_2.html#V1">clockwise_oriented</a> or  <a href="goboard9.html#K1">Rev</a> <font color="Maroon">f</font> is <a href="sprect_2.html#V1">clockwise_oriented</a> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><b>suppose </b></a><a NAME="E1:50_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:50_1_2"/>
( <font color="Maroon">f</font> is <a href="sprect_2.html#V1">clockwise_oriented</a> or  <a href="goboard9.html#K1">Rev</a> <font color="Maroon">f</font> is <a href="sprect_2.html#V1">clockwise_oriented</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">g</font> =  <a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:50_1_2_1"/><i><font color="Green">E39</font></i>: 
 <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T43">SPRECT_2:43</a></i>;<br/>
<a NAME="E2:50_1_2_1"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:50_1_2_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="revrot_1.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_6.html#T98">FINSEQ_6:98</a></i>;<br/>
<a NAME="E4:50_1_2_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="goboard7.html#T38">GOBOARD7:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_2_1_1"/>
(  <a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> is <a href="sprect_2.html#V1">clockwise_oriented</a> or  <a href="goboard9.html#K1">Rev</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> is <a href="sprect_2.html#V1">clockwise_oriented</a> )
 </span><b>by </b><i><a class="txt" href="revrot_1.html#E1:50_1_2"><i><font color="Green">E38</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_1"/>
 <a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> is <a href="sprect_2.html#V1">clockwise_oriented</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">g</font> =  <a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> as  <a href="funct_1.html#V3">V7</a> <a href="goboard5.html#V2">standard</a> <a href="sprect_2.html#V1">clockwise_oriented</a> <a href="goboard5.html#NM1">special_circular_sequence</a> ;<br/><a NAME="E3:50_1_2_1_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:50_1_2_1"><i><font color="Green">E39</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:50_1_2_1_1_1"/>
( <font color="Maroon">f</font> is <a href="sprect_2.html#V1">clockwise_oriented</a> or  <a href="goboard9.html#K1">Rev</a> <font color="Maroon">f</font> is <a href="sprect_2.html#V1">clockwise_oriented</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2"/>
 <a href="goboard9.html#K1">Rev</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> is <a href="sprect_2.html#V1">clockwise_oriented</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">h</font> =  <a href="goboard9.html#K1">Rev</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> as  <a href="funct_1.html#V3">V7</a> <a href="goboard5.html#V2">standard</a> <a href="sprect_2.html#V1">clockwise_oriented</a> <a href="goboard5.html#NM1">special_circular_sequence</a> ;<br/><a NAME="E3:50_1_2_1_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="revrot_1.html#R1" target="_self">just_once_values</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="revrot_1.html#D2" target="_self">REVROT_1:def 2</a><br/>

<a NAME="E1:50_1_2_1_1_2_1"/><i><font color="Green">E43</font></i>: 
 <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T43">SPRECT_2:43</a></i>;<br/>
<a NAME="E2:50_1_2_1_1_2_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_6.html#T46">FINSEQ_6:46</a></i>;<br/>
<a NAME="E3:50_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="revrot_1.html#E4:50_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_6.html#T96">FINSEQ_6:96</a></i>;<br/>
<a NAME="E4:50_1_2_1_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i>, <a class="txt" href="revrot_1.html#E4:50_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_6.html#T98">FINSEQ_6:98</a></i>;<br/>
<b>thus </b><a NAME="E5:50_1_2_1_1_2_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E3:50_1_2_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>

<b>thus </b><i><font color="Green">E61</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="revrot_1.html#E3:50_1_2_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="revrot_1.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E7:50_1_2_1_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E8:50_1_2_1_1_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="revrot_1.html#T2" target="_self">Th2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_2_1_1_2_1_2"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="ref" href="revrot_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_2_1"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_2_1"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E3:50_1_2_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/><a NAME="E3:50_1_2_1_1_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:50_1_2_1_1_2_1"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/><a NAME="E4:50_1_2_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:50_1_2_1_1_2_1_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="revrot_1.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E6:50_1_2_1_1_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i>, , , <a class="ref" href="goboard7.html#T38">GOBOARD7:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2_1_2_2"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2_1_2_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:50_1_2_1_1_2_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="revrot_1.html#E3:50_1_2_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/><a NAME="E3:50_1_2_1_1_2_1_2_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:50_1_2_1_1_2_1"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:50_1_2_1_1_2_1_2_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="revrot_1.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E5:50_1_2_1_1_2_1_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i>, , , <a class="ref" href="goboard7.html#T39">GOBOARD7:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div> <a href="goboard9.html#K1">Rev</a> <font color="Maroon">f</font> = 
 <a href="goboard9.html#K1">Rev</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="revrot_1.html#E1:50_1_2_1"><i><font color="Green">E39</font></i></a>, </i>
<br/>.= 
 <a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">h</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="revrot_1.html#E3:50_1_2_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_6.html#T112">FINSEQ_6:112</a></i>
;<br/><b>hence </b><a NAME="E5:50_1_2_1_1_2"/>
( <font color="Maroon">f</font> is <a href="sprect_2.html#V1">clockwise_oriented</a> or  <a href="goboard9.html#K1">Rev</a> <font color="Maroon">f</font> is <a href="sprect_2.html#V1">clockwise_oriented</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
