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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1"><b>suppose </b></a><a NAME="E1:117_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>assume </b><b>that </b><br/><a NAME="E2:117_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="finseq_6.html#V1" target="_self">circular</a>
 <b>and </b><br/><a NAME="E3:117_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#R2">just_once_values</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E4:117_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E3:117_1_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_4.html#T60">FINSEQ_4:60</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_1_1_1_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> is <a href="realset1.html#V1">trivial</a> or  not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> is <a href="realset1.html#V1">trivial</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:117_1_1_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> is <a href="realset1.html#V1">trivial</a>
 ;<br/><div class="add"><a NAME="E2:117_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:117_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="finseq_6.html#E4:117_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="realset1.html#T14">REALSET1:14</a></i>;<br/><b>hence </b><a NAME="E3:117_1_1_1_1_1"/>
 <a href="finseq_6.html#K1" target="_self">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" target="_self">Rotate</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_6.html#K1" target="_self">Rotate</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_6.html#T99" target="_self">Th99</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:117_1_1_1_1_2"/><i><font color="Green">E103</font></i>: 
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> is <a href="realset1.html#V1">trivial</a>
 ;<br/><div class="add"><a NAME="E2:117_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></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"><font color="Maroon">c<sub>4</sub></font> <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"><font color="Maroon">c<sub>4</sub></font> <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>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><a NAME="E3:117_1_1_1_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <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="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&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"><font color="Maroon">c<sub>4</sub></font> <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_1_1_1_1_2_1"/><i><font color="Green">E105</font></i>: 
not <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
 ;<br/><a NAME="E2:117_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E4:117_1_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="finseq_6.html#E3:117_1_1_1_1_2"><i><font color="Green">E104</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:117_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E4:117_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:117_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E2:117_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_6.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E6:117_1_1_1_1_2_1"/>
not <font color="Maroon">c<sub>4</sub></font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:117_1_1_1_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_6.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E7:117_1_1_1_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="realset1.html#T13">REALSET1:13</a></i>;<br/><a NAME="E8:117_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E107</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:117_1_1_1_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:117_1_1_1_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><b>hence </b><a NAME="E10:117_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:117_1_1_1_1_2_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="finseq_6.html#E5:117_1_1_1_1_2_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="finseq_6.html#E8:117_1_1_1_1_2_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="finseq_6.html#T63" target="_self">Th63</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:117_1_1_1_1_2"/>
 <a href="finseq_6.html#K1" target="_self">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" target="_self">Rotate</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_6.html#K1" target="_self">Rotate</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E3:117_1_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_6.html#T110" target="_self">Th110</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:117_1_1"/>
 <a href="finseq_6.html#K1" target="_self">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" target="_self">Rotate</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_6.html#K1" target="_self">Rotate</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_2"><b>suppose </b></a><a NAME="E1:117_1_2"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:117_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="finseq_6.html#V1" target="_self">circular</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#R2">just_once_values</a> <font color="Maroon">c<sub>2</sub></font> implies  <a href="finseq_6.html#K1" target="_self">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" target="_self">Rotate</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_6.html#K1" target="_self">Rotate</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_6.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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