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<b>let </b><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:52"/><i><font color="Green" title="E34">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c3">f</font> )
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c4">G</font>
 <b>and </b><br/><a NAME="E3:52"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> or <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>suppose </b></a><a NAME="E1:52_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>suppose </b></a><a NAME="E1:52_1_2"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c5">i1</font>, <font color="Maroon" title="c6">j1</font>, <font color="Maroon" title="c7">i2</font>, <font color="Maroon" title="c8">j2</font> being    <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> such that </b><br/><a NAME="E3:52_1_2"/><i><font color="Green" title="E38">A5</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">j1</font> )
 <b>and </b><br/><a NAME="E4:52_1_2"/><i><font color="Green" title="E39">A6</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c8">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c8">j2</font> )
 <b>and </b><br/><a NAME="E5:52_1_2"/><i><font color="Green" title="E40">A7</font></i>: 
( ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> ) or ( <font color="Maroon" title="c5">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> ) or ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> ) or ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:52"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E6:52_1_2"/><i><font color="Green" title="E41">A8</font></i>: 
( <font color="Maroon" title="c5">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i1</font> &amp; <font color="Maroon" title="c7">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">j1</font> &amp; <font color="Maroon" title="c8">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">j2</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c9">lf</font> =  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>;<br/><b>set </b><font color="Maroon" title="c10">lfn</font> =  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>;<br/><b>set </b><font color="Maroon" title="c11">rf</font> =  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>;<br/><b>set </b><font color="Maroon" title="c12">rfn</font> =  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>;<br/><a NAME="E7:52_1_2"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c4">G</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E2:52"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="goboard1.html#T38" title="GOBOARD1:th.38">GOBOARD1:38</a></i>;<br/><a NAME="E8:52_1_2"/><i><font color="Green" title="E43">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:52_1_2"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><a NAME="E9:52_1_2"/><b>then </b>
( <font color="Maroon" title="c1">k</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="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E3:52"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/><a NAME="E10:52_1_2"/><b>then </b><i><font color="Green" title="E44">A11</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_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="E1:52_1_2_1_1"/>
( ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> ) or ( <font color="Maroon" title="c5">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> ) or ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> ) or ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="gobrd13.html#E5:52_1_2"><i><font color="Green" title="E40">A7</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_1"/><i><font color="Green" title="E45">A12</font></i>: 
( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> )
 ;<br/><div class="add"><b>hence </b> <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c8">j2</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:52"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="gobrd13.html#D5" target="_self" title="GOBRD13:def.5">Def5</a></i>
<br/>.= 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E3:52"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E7:52_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="gobrd13.html#E8:52_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="gobrd13.html#E10:52_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="gobrd13.html#D5" target="_self" title="GOBRD13:def.5">Def5</a></i>
;<br/><br/><b>thus </b> <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c8">j2</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:52"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="gobrd13.html#D4" target="_self" title="GOBRD13:def.4">Def4</a></i>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E3:52"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E7:52_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="gobrd13.html#E8:52_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="gobrd13.html#E10:52_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="gobrd13.html#D4" target="_self" title="GOBRD13:def.4">Def4</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2"/><i><font color="Green" title="E45">A13</font></i>: 
( <font color="Maroon" title="c5">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> )
 ;<br/><div class="add"><b>hence </b> <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c8">j2</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:52"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="gobrd13.html#D5" target="_self" title="GOBRD13:def.5">Def5</a></i>
<br/>.= 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E3:52"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E7:52_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="gobrd13.html#E8:52_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="gobrd13.html#E10:52_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_2"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="gobrd13.html#D5" target="_self" title="GOBRD13:def.5">Def5</a></i>
;<br/><br/><b>thus </b> <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c7">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:52"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_2"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="gobrd13.html#D4" target="_self" title="GOBRD13:def.4">Def4</a></i>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E3:52"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E7:52_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="gobrd13.html#E8:52_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="gobrd13.html#E10:52_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_2"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="gobrd13.html#D4" target="_self" title="GOBRD13:def.4">Def4</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_3"/><i><font color="Green" title="E45">A14</font></i>: 
( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> )
 ;<br/><div class="add"><b>hence </b> <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:52"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="gobrd13.html#D5" target="_self" title="GOBRD13:def.5">Def5</a></i>
<br/>.= 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E3:52"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E7:52_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="gobrd13.html#E8:52_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="gobrd13.html#E10:52_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_3"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="gobrd13.html#D5" target="_self" title="GOBRD13:def.5">Def5</a></i>
;<br/><br/><b>thus </b> <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c8">j2</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:52"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_3"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="gobrd13.html#D4" target="_self" title="GOBRD13:def.4">Def4</a></i>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E3:52"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E7:52_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="gobrd13.html#E8:52_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="gobrd13.html#E10:52_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_3"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="gobrd13.html#D4" target="_self" title="GOBRD13:def.4">Def4</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_4"/><i><font color="Green" title="E45">A15</font></i>: 
( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><div class="add"><b>hence </b> <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c7">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:52"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="gobrd13.html#D5" target="_self" title="GOBRD13:def.5">Def5</a></i>
<br/>.= 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E3:52"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E7:52_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="gobrd13.html#E8:52_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="gobrd13.html#E10:52_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_4"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="gobrd13.html#D5" target="_self" title="GOBRD13:def.5">Def5</a></i>
;<br/><br/><b>thus </b> <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:52"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_4"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="gobrd13.html#D4" target="_self" title="GOBRD13:def.4">Def4</a></i>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<b>by </b><i><a class="txt" href="gobrd13.html#E1:52"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="gobrd13.html#E3:52"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="gobrd13.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="gobrd13.html#E4:52_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="gobrd13.html#E6:52_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="gobrd13.html#E7:52_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="gobrd13.html#E8:52_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="gobrd13.html#E10:52_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="gobrd13.html#E1:52_1_2_1_1_4"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="gobrd13.html#D4" target="_self" title="GOBRD13:def.4">Def4</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:52_1_2"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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