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<b>let </b><font color="Maroon">f</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon">n</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:14"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">fs1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">fs2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:14"/><i><font color="Green">E42</font></i>: 
( ( for <font color="Olive">k</font>, <font color="Olive">k1</font>, <font color="Olive">k2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs1</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/>( <font color="Olive">k1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Olive">k2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> ) ) &amp; <font color="Maroon">fs1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">fs1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">fs2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">fs2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font> holds <br/><font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font>,<font color="Maroon">fs2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font>,<font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">fs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd10.html#R2">are_adjacent2</a>  ) )
 <b>by </b><i>, , <a class="ref" href="gobrd10.html#T8">GOBRD10:8</a></i>;<br/>
<a NAME="E6:14"/><i><font color="Green">E48</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E7:14"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:14"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs2</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:14"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E10:14"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i2</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E11:14"/><b>then </b><i><font color="Green">E52</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs2</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E12:14"/><b>then </b>
( <font color="Maroon">fs1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">fs2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E13:14"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , , , , </i>;<br/>


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