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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/><b>let </b><font color="Maroon" title="c2">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="c3">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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">k</font> be    <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> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:55"/><i><font color="Green" title="E35">A1</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font>
 <b>and </b><br/><a NAME="E2:55"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E3:55"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<a NAME="E4:55"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D1" title="FINSEQ_5:def.1">FINSEQ_5:def 1</a></i>;<br/>
<a NAME="E5:55"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c4">k</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> <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E2:55"><i><font color="Green" title="E36">A2</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="55_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1"/>
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> or <font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span><b>by </b><i><a class="ref" href="topreal8.html#T4" title="TOPREAL8:th.4">TOPREAL8:4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1"><b>suppose </b></a><a NAME="E1:55_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 ;<br/><div class="add"><a NAME="E2:55_1_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</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="c2">f</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/><a NAME="E3:55_1_1"/><b>then </b>
<font color="Maroon" title="c4">k</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="c2">f</font>
 <b>by </b><i><a class="txt" href="jordan1j.html#E5:55"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:55_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">G</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="jordan1j.html#E1:55"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="jordan1j.html#E3:55"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="jordan1j.html#E4:55"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="jordan1j.html#E5:55"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2"><b>suppose </b></a><a NAME="E1:55_1_2"/>
<font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_2"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">G</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="jordan1j.html#E2:55"><i><font color="Green" title="E36">A2</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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