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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">i1</font>, <font color="Maroon" title="c3">j1</font>, <font color="Maroon" title="c4">i2</font>, <font color="Maroon" title="c5">j2</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><a NAME="E1:11"/><i><font color="Green" title="E13">A1</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c3">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c5">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font>,<font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font> <a href="gobrd10.html#R2" title="GOBRD10:pred.2">are_adjacent2</a>  )
 ;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green" title="E14">A2</font></i>: 
( ( <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c4">i2</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  &amp; <font color="Maroon" title="c3">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j2</font> ) or ( <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c3">j1</font>,<font color="Maroon" title="c5">j2</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  ) )
 
<b>by </b><i><a class="ref" href="gobrd10.html#D2" title="GOBRD10:def.2">GOBRD10:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1"/>
( ( ( <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) &amp; <font color="Maroon" title="c3">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j2</font> ) or ( <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> &amp; ( <font color="Maroon" title="c5">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c3">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) ) )
 </span><b>by </b><i><a class="txt" href="gobrd12.html#E2:11"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="gobrd10.html#D1" title="GOBRD10:def.1">GOBRD10:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1"><b>case </b></a><a NAME="E1:11_1_1_1"/>
( ( <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) &amp; <font color="Maroon" title="c3">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_1"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span> iff  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd12.html#E1:11"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="gobrd12.html#E11"><i><font color="Green" title="E10">Lm5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_2"><b>case </b></a><a NAME="E1:11_1_1_2"/>
( <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> &amp; ( <font color="Maroon" title="c5">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c3">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_2"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span> iff  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd12.html#E1:11"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="gobrd12.html#E12"><i><font color="Green" title="E11">Lm6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:11"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span> iff  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 ;<br/>


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