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<div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:102"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> is <a href="finseq_6.html#V1" target="_self">circular</a>
 <b>and </b><br/><a NAME="E2:102"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E3:102"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_6.html#K1" target="_self">Rotate</a> <font color="Maroon">f</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_6.html#T36" target="_self">Th36</a>, <a class="ref" href="finseq_6.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E4:102"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E5:102"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</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#T36" target="_self">Th36</a>, <a class="ref" href="finseq_5.html#T58">FINSEQ_5:58</a></i>;<br/>
<a NAME="E6:102"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T51">FINSEQ_5:51</a></i>;<br/>
<a NAME="E7:102"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T36">FINSEQ_5:36</a></i>;<br/>
<a NAME="E8:102"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="tarski.html#R1">c=</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#T40" target="_self">Th40</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E9:102"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" target="_self">Rotate</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</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#T37" target="_self">Th37</a>, <a class="ref" href="finseq_6.html#T38" target="_self">Th38</a>, <a class="ref" href="finseq_6.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E10:102"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:102"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E13:102"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E14:102"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_6.html#T41" target="_self">Th41</a>, <a class="ref" href="finseq_6.html#T42" target="_self">Th42</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or ( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 ) or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1"><b>suppose </b></a><a NAME="E1:102_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E41</font></i>: <font color="Maroon">x</font> = 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_6.html#T34" target="_self">Th34</a>, , <a class="ref" href="finseq_6.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_6.html#T36" target="_self">Th36</a>, <a class="ref" href="finseq_5.html#T57">FINSEQ_5:57</a></i>
;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D2">FINSEQ_5:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><a NAME="E4:102_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:102_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:102_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E7:102_1_1"/>
<font color="Maroon">x</font> <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" target="_self">Rotate</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T37" target="_self">Th37</a>, <a class="ref" href="finseq_6.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:102_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:102_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E3:102_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_6.html#T41" target="_self">Th41</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:102_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_6.html#T44" target="_self">Th44</a>, <a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><b>consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:102_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_6.html#T45" target="_self">Th45</a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E7:102_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T36" target="_self">Th36</a>, , <a class="ref" href="finseq_5.html#T45">FINSEQ_5:45</a></i>;<br/><a NAME="E8:102_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T46" target="_self">Th46</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:102_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/><a NAME="E10:102_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><a NAME="E11:102_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T47" target="_self">Th47</a>, <a class="ref" href="finseq_6.html#T48" target="_self">Th48</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:102_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="finseq_6.html#T46" target="_self">Th46</a>, <a class="ref" href="finseq_6.html#T47" target="_self">Th47</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E13:102_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T49" target="_self">Th49</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:102_1_2"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E15:102_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_6.html#T46" target="_self">Th46</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E16:102_1_2"/>
not <font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="finseq_6.html#T36" target="_self">Th36</a>, <a class="ref" href="finseq_5.html#T50">FINSEQ_5:50</a></i>;<br/><a NAME="E17:102_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T32">FINSEQ_5:32</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_6.html#T47" target="_self">Th47</a>, <a class="ref" href="finseq_6.html#T50" target="_self">Th50</a>, <a class="ref" href="finseq_6.html#T51" target="_self">Th51</a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_6.html#T36" target="_self">Th36</a>, <a class="ref" href="finseq_6.html#T52" target="_self">Th52</a>, <a class="ref" href="finseq_5.html#T46">FINSEQ_5:46</a></i>
;<br/><a NAME="E19:102_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">p</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_6.html#T50" target="_self">Th50</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E20:102_1_2"/>
<font color="Maroon">x</font> <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" target="_self">Rotate</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T37" target="_self">Th37</a>, <a class="ref" href="finseq_6.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_3"><b>suppose </b></a><a NAME="E1:102_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:102_1_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:102_1_3"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_6.html#T36" target="_self">Th36</a>, <a class="ref" href="finseq_5.html#T53">FINSEQ_5:53</a></i>;<br/><a NAME="E5:102_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="finseq_6.html#T41" target="_self">Th41</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:102_1_3"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:102_1_3"/><b>then </b>
<font color="Maroon">j</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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T44" target="_self">Th44</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:102_1_3"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:102_1_3"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E10:102_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T36" target="_self">Th36</a>, <a class="ref" href="finseq_6.html#T45" target="_self">Th45</a>, <a class="ref" href="finseq_5.html#T55">FINSEQ_5:55</a></i>;<br/><a NAME="E11:102_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_6.html#T45" target="_self">Th45</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E12:102_1_3"/>
<font color="Maroon">x</font> <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" target="_self">Rotate</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#T37" target="_self">Th37</a>, <a class="ref" href="finseq_6.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
