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

<a NAME="E1:49_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></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/>
<a NAME="E2:49_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="revrot_1.html#T33" target="_self">Th33</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_1"/><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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_1"/><b>then </b>
 <a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:49_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="revrot_1.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_1_1"/>
<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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<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> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K24">N-most</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E2:49_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="revrot_1.html#E1:49_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="sprect_2.html#T34">SPRECT_2:34</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="sprect_2.html#D4">SPRECT_2:def 4</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="revrot_1.html#R1" target="_self">just_once_values</a>  <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">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font>
; <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="E1:49_1_1_1_2_1"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="jordan5d.html#D15">JORDAN5D:def 15</a></i>;<br/>
<a NAME="E2:49_1_1_1_2_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E3:49_1_1_1_2_1"/><i><font color="Green">E42</font></i>: 
 <a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>

<b>thus </b><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">c<sub>2</sub></font></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="jordan5d.html#D15">JORDAN5D:def 15</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="revrot_1.html#E3:49_1_1_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E5:49_1_1_1_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>assume </b><a NAME="E7:49_1_1_1_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_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="E1:49_1_1_1_2_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="revrot_1.html#E7:49_1_1_1_2_1"><i><font color="Green">E45</font></i></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="49_1_1_1_2_1_2_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_2_1_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_2_1_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="revrot_1.html#E5:49_1_1_1_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_1_2_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E2:49_1_1_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="revrot_1.html#E4:49_1_1_1_2_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="revrot_1.html#E1:49_1_1_1_2_1_2_1"><i><font color="Green">E46</font></i></a>, <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="49_1_1_1_2_1_2_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_2_1_2_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:49_1_1_1_2_1_2_2"/><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan5d.html#K15">n_w_n</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="revrot_1.html#E1:49_1_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="revrot_1.html#E1:49_1_1_1_2_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="revrot_1.html#E4:49_1_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="E3:49_1_1_1_2_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="revrot_1.html#E5:49_1_1_1_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E4:49_1_1_1_2_1_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E4:49_1_1_1_2_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="revrot_1.html#E1:49_1_1_1_2_1_2_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="revrot_1.html#E2:49_1_1_1_2_1_2_2"><i><font color="Green">E47</font></i></a>, <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 NAME="E3:49_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K24">N-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_2.html#D4">SPRECT_2:def 4</a></i>;<br/><b>hence </b><a NAME="E4:49_1_1_1_2"/>
<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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<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> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K24">N-most</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="revrot_1.html#E2:49_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="revrot_1.html#E2:49_1_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_6.html#T111">FINSEQ_6:111</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="sprect_2.html#D4">SPRECT_2:def 4</a><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
