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<b>let </b><font color="Maroon">z</font> be  non <a href="funct_1.html#V3">constant</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/>

<b>set </b><font color="Maroon">g</font> =  <a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:93"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">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">z</font> holds <br/><font color="Maroon">z</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">z</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="E2:93"/><i><font color="Green">E32</font></i>: 
 <a href="pscomp_1.html#K33">S-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T46">SPRECT_2:46</a></i>;<br/>
<a NAME="E3:93"/><i><font color="Green">E34</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">z</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">z</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="revrot_1.html#T33">REVROT_1:33</a></i>;<br/>
<b>assume </b><a NAME="E4:93"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K34">S-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span>
 ;<br/>

<a NAME="E5:93"/><i><font color="Green">E36</font></i>: 
 <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">z</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="sprect_5.html#T12" target="_self">Th12</a>, <a class="ref" href="revrot_1.html#T16">REVROT_1:16</a></i>;<br/>
<a NAME="E6:93"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">z</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</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#K33">S-max</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">z</font>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_5.html#T10" target="_self">Th10</a>, <a class="ref" href="sprect_5.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_6.html#T98">FINSEQ_6:98</a></i>;<br/>
<a NAME="E7:93"/><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> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">z</font>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <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">z</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T43">SPRECT_2:43</a></i>;<br/>
<a NAME="E8:93"/><i><font color="Green">E39</font></i>: 
 <a href="pscomp_1.html#K30">N-max</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">z</font>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <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">z</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T44">SPRECT_2:44</a></i>;<br/>
<a NAME="E9:93"/><i><font color="Green">E40</font></i>: 
 <a href="pscomp_1.html#K34">S-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">z</font>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <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">z</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T45">SPRECT_2:45</a></i>;<br/>
<a NAME="E10:93"/><i><font color="Green">E41</font></i>: 
<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> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">z</font>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span></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">z</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">z</font>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span></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">z</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_5.html#E23"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="sprect_5.html#E10:93"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E11:93"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">z</font>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span></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">z</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <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> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">z</font>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span></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">z</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_5.html#E23"><i><font color="Green">Lemma65</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E12:93"/>
<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">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_5.html#T11" target="_self">Th11</a>, <a class="txt" href="sprect_5.html#E22"><i><font color="Green">Lemma64</font></i></a>, <a class="txt" href="sprect_5.html#E24"><i><font color="Green">Lemma66</font></i></a>, <a class="txt" href="sprect_5.html#E25"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="sprect_5.html#E26"><i><font color="Green">Lemma68</font></i></a>, <a class="txt" href="sprect_5.html#E27"><i><font color="Green">Lemma69</font></i></a>, </i>;<br/>


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