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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="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a>;<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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:54"/><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="c4">k</font>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green" title="E35">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#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E3:54"/><i><font color="Green" title="E36">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>
 <b>and </b><br/><a NAME="E4:54"/><i><font color="Green" title="E37">A4</font></i>: 
<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/>

<a NAME="E5:54"/>
<font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:54"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="jordan1j.html#E2:54"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then </b><i><font color="Green" title="E38">A5</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#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></span>)</span> = 
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">f</font>,1,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
<br/>.= 
 <a href="jordan3.html#K4" title="JORDAN3:func.4">R_Cut</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="txt" href="jordan1j.html#E4:54"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="jordan1g.html#T57" title="JORDAN1G:th.57">JORDAN1G:57</a></i>
;<br/>
<a NAME="E7:54"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c4">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="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E2:54"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:54"/>
<font color="Maroon" title="c3">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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="txt" href="jordan1j.html#E4:54"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/>
<a NAME="E9:54"/><b>then </b>
<font color="Maroon" title="c4">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="c2">f</font>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E7:54"><i><font color="Green" title="E39">A6</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="E10:54"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4" title="JORDAN3:func.4">R_Cut</a> <font color="Maroon" title="c2">f</font>,<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"><a href="jordan3.html#K4" title="JORDAN3:func.4">R_Cut</a> <font color="Maroon" title="c2">f</font>,<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#E1:54"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="jordan1j.html#E3:54"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="jordan1j.html#E6:54"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="jordan1j.html#E7:54"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/>


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