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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">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>assume </b><a NAME="E1:10"/><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> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="finseq_6.html#K1">Rotate</a> <font color="Maroon">f</font>,<font color="Maroon">q</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">q</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">q</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_6.html#D2">FINSEQ_6:def 2</a></i>;<br/>
<b>assume </b><a NAME="E3:10"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E4:10"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">p</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#K1">-:</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_5.html#T49">FINSEQ_5:49</a></i>;<br/>
<a NAME="E5:10"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/>
<a NAME="E6:10"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1"/>
( <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i>, , <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="10_1_1"><b>suppose </b></a><a NAME="E1:10_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:10_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_5.html#T41">FINSEQ_5:41</a></i>;<br/><b>hence </b><a NAME="E3:10_1_1"/>
<font color="Maroon">p</font> <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>,<font color="Maroon">q</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="nat_1.html#K1">+</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#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2"><b>suppose </b></a><a NAME="E1:10_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:10_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E3:10_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 1 <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> <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">q</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_6.html#T61">FINSEQ_6:61</a></i>;<br/><a NAME="E4:10_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> 1 is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i/>;<br/><a NAME="E5:10_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">p</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">q</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_6.html#T62">FINSEQ_6:62</a></i>;<br/><a NAME="E6:10_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</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="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_6.html#T65">FINSEQ_6:65</a></i>;<br/><a NAME="E7:10_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">q</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="rfinseq.html#K1">/^</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_6.html#T83">FINSEQ_6:83</a></i>;<br/><a NAME="E8:10_1_2"/><b>then </b><i><font color="Green">E57</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="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#R1">misses</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="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T72">FINSEQ_4:72</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">q</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">q</font></span><a href="binarith.html#K11">*&gt;</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_5.html#T27">FINSEQ_5:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="finseq_5.html#D1">FINSEQ_5:def 1</a></i>
;<br/><a NAME="E10:10_1_2"/><b>then </b><i><font color="Green">E58</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="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_5.html#K1">-:</a> <font color="Maroon">q</font></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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">q</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"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">q</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><a NAME="E11:10_1_2"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E12:10_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">q</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E13:10_1_2"/><b>then </b>
<font color="Maroon">p</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="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E14:10_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
not <font color="Maroon">p</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="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E15:10_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">q</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_6.html#T45">FINSEQ_6:45</a></i>;<br/><a NAME="E16:10_1_2"/><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="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">q</font></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"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">q</font></span><a href="binarith.html#K11">*&gt;</a></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="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><a NAME="E17:10_1_2"/><b>then </b>
not <font color="Maroon">p</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="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_5.html#K2">:-</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E18:10_1_2"/><b>then </b>
not <font color="Maroon">p</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">q</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_6.html#T89">FINSEQ_6:89</a></i>;<br/><a NAME="E19:10_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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">q</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</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">q</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><font color="Maroon">p</font> <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>,<font color="Maroon">q</font></span>)</span> = 
<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">q</font></span>)</span></span>)</span> <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> <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">q</font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_6.html#T9">FINSEQ_6:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">q</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</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><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> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_5.html#T53">FINSEQ_5:53</a></i>
<br/>.= 
<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="nat_1.html#K1">+</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#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span>

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

</div>
