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<div class="add">

<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>assume </b><a NAME="E1:36"/><i><font color="Green" title="E31">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">i1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:36"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c2">i1</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> <font color="Maroon" title="c1">f</font>
 <b>and </b><br/><a NAME="E4:36"/><i><font color="Green" title="E33">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="goboard7.html#T32" target="_self" title="GOBOARD7:th.32">Th32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:36"/><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c3">i2</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> <font color="Maroon" title="c1">f</font>
 <b>and </b><br/><a NAME="E7:36"/><i><font color="Green" title="E35">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="ref" href="goboard7.html#T33" target="_self" title="GOBOARD7:th.33">Th33</a></i>;<br/>
<a NAME="E8:36"/><i><font color="Green" title="E36">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="goboard7.html#E32"><i><font color="Green" title="E26">Lm2</font></i></a></i>;<br/>
<a NAME="E9:36"/><b>then </b><i><font color="Green" title="E37">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E10:36"/><b>then </b><i><font color="Green" title="E38">A8</font></i>: 
( 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> &amp; 2 <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="goboard7.html#E8:36"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 </span><b>by </b><i><a class="txt" href="goboard7.html#E9:36"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><b>suppose </b></a><a NAME="E1:36_1_1"/><i><font color="Green" title="E39">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:36_1_1"/><i><font color="Green" title="E40">A10</font></i>: 
<font color="Maroon" title="c2">i1</font> <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="txt" href="goboard7.html#E3:36"><i><font color="Green" title="E32">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="E3:36_1_1"/><i><font color="Green" title="E41">A11</font></i>: 
<font color="Maroon" title="c2">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="goboard7.html#E3:36"><i><font color="Green" title="E32">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="E4:36_1_1"/><i><font color="Green" title="E42">A12</font></i>: 
<font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4
 <b>by </b><i><a class="txt" href="goboard7.html#E1:36"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="goboard7.html#E2:36_1_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:36_1_1"/><i><font color="Green" title="E43">A13</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_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="E1:36_1_1_1_1"/>
( <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="goboard7.html#E4:36"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="goboard7.html#E1:36_1_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="goboard7.html#E3:36_1_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="goboard7.html#E4:36_1_1"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1_1_1"/><i><font color="Green" title="E44">A14</font></i>: 
<font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E45">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 <b>by </b><i><a class="txt" href="goboard7.html#E3:36"><i><font color="Green" title="E32">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="E3:36_1_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3
 <b>by </b><i><a class="txt" href="goboard7.html#E4:36"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="goboard7.html#E5:36_1_1"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="goboard7.html#E1:36_1_1_1_1_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:36_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E46">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:36_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E47">A17</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <b>by </b><i><a class="txt" href="goboard7.html#E1:36"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div><i><font color="Green" title="E48">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><b>then </b><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="txt" href="goboard7.html#E1:36_1_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
;<br/><b>hence </b><a NAME="E3:36_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard7.html#E10:36"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="goboard7.html#T31" target="_self" title="GOBOARD7:th.31">Th31</a></i>;<br/></div><b>end;</b></div><a NAME="E7:36_1_1_1_1_1"/><i><font color="Green" title="E49">A19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard7.html#E4:36"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="goboard7.html#E1:36_1_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="goboard7.html#E1:36_1_1_1_1_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="goboard7.html#E2:36_1_1_1_1_1"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/><a NAME="E8:36_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard7.html#E4:36_1_1_1_1_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/><b>hence </b><a NAME="E9:36_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard7.html#E4:36"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="goboard7.html#E1:36_1_1_1_1_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="goboard7.html#E5:36_1_1_1_1_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="goboard7.html#E6:36_1_1_1_1_1"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="goboard7.html#E7:36_1_1_1_1_1"><i><font color="Green" title="E49">A19</font></i></a>, <a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_1_1_2"/>
<font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard7.html#E1:36"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="goboard7.html#E4:36"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="goboard7.html#E2:36_1_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="goboard7.html#E5:36_1_1"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:36_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2"><b>suppose </b></a><a NAME="E1:36_1_2"/><i><font color="Green" title="E39">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:36_1_2"/><i><font color="Green" title="E40">A21</font></i>: 
<font color="Maroon" title="c3">i2</font> <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="txt" href="goboard7.html#E6:36"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:36_1_2"/><i><font color="Green" title="E41">A22</font></i>: 
<font color="Maroon" title="c3">i2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="goboard7.html#E6:36"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:36_1_2"/><i><font color="Green" title="E42">A23</font></i>: 
<font color="Maroon" title="c3">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4
 <b>by </b><i><a class="txt" href="goboard7.html#E1:36"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="goboard7.html#E2:36_1_2"><i><font color="Green" title="E40">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:36_1_2"/><i><font color="Green" title="E43">A24</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_2_1_1"/>
( <font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="goboard7.html#E7:36"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="goboard7.html#E1:36_1_2"><i><font color="Green" title="E39">A20</font></i></a>, <a class="txt" href="goboard7.html#E3:36_1_2"><i><font color="Green" title="E41">A22</font></i></a>, <a class="txt" href="goboard7.html#E4:36_1_2"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_2_1_1_1"/><i><font color="Green" title="E44">A25</font></i>: 
<font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:36_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E45">A26</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 <b>by </b><i><a class="txt" href="goboard7.html#E6:36"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:36_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3
 <b>by </b><i><a class="txt" href="goboard7.html#E7:36"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="goboard7.html#E5:36_1_2"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="goboard7.html#E1:36_1_2_1_1_1"><i><font color="Green" title="E44">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:36_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E46">A27</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:36_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E47">A28</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <b>by </b><i><a class="txt" href="goboard7.html#E1:36"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div><i><font color="Green" title="E48">A29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><b>then </b><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="txt" href="goboard7.html#E1:36_1_2"><i><font color="Green" title="E39">A20</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
;<br/><b>hence </b><a NAME="E3:36_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard7.html#E10:36"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="goboard7.html#T31" target="_self" title="GOBOARD7:th.31">Th31</a></i>;<br/></div><b>end;</b></div><a NAME="E7:36_1_2_1_1_1"/><i><font color="Green" title="E49">A30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard7.html#E7:36"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="goboard7.html#E1:36_1_2"><i><font color="Green" title="E39">A20</font></i></a>, <a class="txt" href="goboard7.html#E1:36_1_2_1_1_1"><i><font color="Green" title="E44">A25</font></i></a>, <a class="txt" href="goboard7.html#E2:36_1_2_1_1_1"><i><font color="Green" title="E45">A26</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/><a NAME="E8:36_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="goboard7.html#E4:36_1_2_1_1_1"><i><font color="Green" title="E46">A27</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/><b>hence </b><a NAME="E9:36_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard7.html#E7:36"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="goboard7.html#E1:36_1_2_1_1_1"><i><font color="Green" title="E44">A25</font></i></a>, <a class="txt" href="goboard7.html#E5:36_1_2_1_1_1"><i><font color="Green" title="E47">A28</font></i></a>, <a class="txt" href="goboard7.html#E6:36_1_2_1_1_1"><i><font color="Green" title="E48">A29</font></i></a>, <a class="txt" href="goboard7.html#E7:36_1_2_1_1_1"><i><font color="Green" title="E49">A30</font></i></a>, <a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_2_1_1_2"/>
<font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard7.html#E1:36"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="goboard7.html#E7:36"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="goboard7.html#E2:36_1_2"><i><font color="Green" title="E40">A21</font></i></a>, <a class="txt" href="goboard7.html#E5:36_1_2"><i><font color="Green" title="E43">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:36_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
