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<b>let </b><font color="Maroon">f</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E32</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <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">f</font> )
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green">E33</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:43"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:43"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:43"/><b>then </b>
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E6:43"/><b>then </b>
<font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 
<b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:43"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 
<b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<i><font color="Green">E43</font></i>: <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan4.html#T21" target="_self">Th21</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<a NAME="E9:43"/>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>;<br/>
<a NAME="E10:43"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan4.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan3.html#T21">JORDAN3:21</a></i>;<br/>
<a NAME="E11:43"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 
<b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, , <a class="ref" href="jordan4.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E12:43"/>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan4.html#T20" target="_self">Th20</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E13:43"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan4.html#T19" target="_self">Th19</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E14:43"/>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 <b>by </b><i><a class="ref" href="jordan4.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan4.html#T24" target="_self">Th24</a>, </i>;<br/>


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