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<b>let </b><font color="Maroon" title="c1">f</font> be  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E3:8"/><i><font color="Green" title="E10">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">i</font> = <font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c5">j</font> = <font color="Maroon" title="c3">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="sprect_5.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="sprect_5.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E5:8"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x2</font>
 ;<br/>

<a NAME="E6:8"/><i><font color="Green" title="E12">A5</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="txt" href="sprect_5.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:8"/><i><font color="Green" title="E13">A6</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 
<b>by </b><i><a class="txt" href="sprect_5.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:8"/>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="sprect_5.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="finseq_5.html#T29" title="FINSEQ_5:th.29">FINSEQ_5:29</a></i>;<br/>
<a NAME="E9:8"/><b>then </b><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:8"/><i><font color="Green" title="E15">A8</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="sprect_5.html#E6:8"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E11:8"/>
<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="sprect_5.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="finseq_5.html#T29" title="FINSEQ_5:th.29">FINSEQ_5:29</a></i>;<br/>
<a NAME="E12:8"/><b>then </b><i><font color="Green" title="E16">A9</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E13:8"/><i><font color="Green" title="E17">A10</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="sprect_5.html#E7:8"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> or <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> )
 </span><b>by </b><i><a class="txt" href="sprect_5.html#E5:8"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>suppose </b></a><a NAME="E1:8_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1"/><b>then </b><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="sprect_5.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="sprect_5.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="sprect_5.html#E3:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="sprect_5.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="sprect_5.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/><b>hence </b><a NAME="E4:8_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_5.html#E10:8"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="sprect_5.html#E12:8"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="sprect_5.html#E2:8_1_1"><i><font color="Green" title="E18">A11</font></i></a>, <a class="ref" href="goboard7.html#T39" title="GOBOARD7:th.39">GOBOARD7:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>suppose </b></a><a NAME="E1:8_1_2"/>
<font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font>
 ;<br/><div class="add"><a NAME="E2:8_1_2"/><b>then </b><i><font color="Green" title="E18">A12</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="sprect_5.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="sprect_5.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="sprect_5.html#E3:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="sprect_5.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="sprect_5.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/><b>hence </b><a NAME="E4:8_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_5.html#E9:8"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="sprect_5.html#E13:8"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="sprect_5.html#E2:8_1_2"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="goboard7.html#T39" title="GOBOARD7:th.39">GOBOARD7:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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