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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">F1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">F2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green">E41</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &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>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></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>and </b><br/><a NAME="E3:12"/>
for <font color="Olive">i</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">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> holds <br/><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font>,<font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font>,<font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd10.html#R2">are_adjacent2</a> 
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">i</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">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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> )
 ;<br/>

<b>reconsider </b><font color="Maroon">F</font> = <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> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<b>consider </b><font color="Maroon">i1</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="E7:12"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &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>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></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>reconsider </b><font color="Maroon">kw1</font> = <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font>, <font color="Maroon">kw2</font> = <font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">kw1</font> <a href="nat_1.html#K1">+</a> 1, <font color="Maroon">k2</font> = <font color="Maroon">kw2</font> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E10:12"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E11:12"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:12"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:12"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <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>,<span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E14:12"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/>for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">x1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">x2</font>] holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
;<br/>
<a NAME="E15:12"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">x</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">x</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E16:12"/>
 ex <font color="Olive">Mm</font> being   <a href="matrix_1.html#M1">Matrix</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1, <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span> st <br/>for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Olive">Mm</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">Mm</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>]
 
<b>from </b><i><a class="ref" href="matrix_1.html#S2">MATRIX_1:sch 2</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">Mm</font> being    <a href="matrix_1.html#M1">Matrix</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1, <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E18:12"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">Mm</font> holds <br/><font color="Maroon">Mm</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <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>,<span class="p2">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 ;<br/>
<a NAME="E19:12"/>
( <font color="Maroon">kw1</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">kw2</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> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E20:12"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<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/>
<a NAME="E21:12"/>
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E22:12"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E23:12"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Mm</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E24:12"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Mm</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E25:12"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">Mm</font> is    <a href="matrix_1.html#M1">Matrix</a> of  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Mm</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1, <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E26:12"/><i><font color="Green">E63</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Mm</font>
 
<b>by </b><i/>;<br/>
<a NAME="E27:12"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">Mm</font>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/>
<a NAME="E28:12"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Mm</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/>
<a NAME="E29:12"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">k2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> )
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E30:12"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">k1</font>,<font color="Maroon">k2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">Mm</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Mm</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E31:12"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">k1</font>,<font color="Maroon">k2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">Mm</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E32:12"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">Mm</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font> <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/>
<a NAME="E33:12"/>
for <font color="Olive">i1</font>, <font color="Olive">j1</font>, <font color="Olive">i2</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">i1</font>,<font color="Olive">j1</font>,<font color="Olive">i2</font>,<font color="Olive">j2</font> <a href="gobrd10.html#R2">are_adjacent2</a>  holds <br/>( <font color="Maroon">Mm</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i1</font>,<font color="Olive">j1</font> <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> iff <font color="Maroon">Mm</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<font color="Olive">j2</font> <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1</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">j1</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">i2</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">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">i1</font>,<font color="Maroon">j1</font>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> <a href="gobrd10.html#R2">are_adjacent2</a>  )
 ;<br/>

<b>reconsider </b><font color="Maroon">ii1</font> = <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1, <font color="Maroon">ii2</font> = <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1, <font color="Maroon">jj1</font> = <font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1, <font color="Maroon">jj2</font> = <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:12_1"/><i><font color="Green">E70</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="gobrd12.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:12_1"/><i><font color="Green">E71</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="gobrd12.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:12_1"/><i><font color="Green">E72</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="gobrd12.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:12_1"/><i><font color="Green">E73</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="gobrd12.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:12_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="gobrd12.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E8:12_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E9:12_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="gobrd12.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E10:12_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="gobrd12.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E11:12_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E12:12_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="gobrd12.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E13:12_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="gobrd12.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E14:12_1"/><b>then </b>
<font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E15:12_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="gobrd12.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E16:12_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="gobrd12.html#E1:12"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E17:12_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E18:12_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="gobrd12.html#E1:12"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E19:12_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">ii1</font>,<font color="Maroon">jj1</font>,<font color="Maroon">ii2</font>,<font color="Maroon">jj2</font> <a href="gobrd10.html#R2">are_adjacent2</a> 
 
<b>by </b><i><a class="txt" href="gobrd12.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="gobrd12.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="gobrd12.html#T2" target="_self">Th2</a>, <a class="ref" href="gobrd12.html#T3" target="_self">Th3</a>, <a class="txt" href="gobrd12.html#E1:12"><i><font color="Green">E37</font></i></a>, <a class="ref" href="gobrd10.html#T4">GOBRD10:4</a></i>;<br/>
<a NAME="E20:12_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">Mm</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Mm</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , <a class="txt" href="gobrd12.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E21:12_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">Mm</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E22:12_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">Mm</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <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>,<span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E23:12_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">Mm</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Mm</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , <a class="txt" href="gobrd12.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E24:12_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">Mm</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E25:12_1"/><b>then </b>
<font color="Maroon">Mm</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <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>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E26:12_1"/>
( <font color="Maroon">Mm</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> <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> iff <font color="Maroon">Mm</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> <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>, , <a class="ref" href="gobrd12.html#T4" target="_self">Th4</a>, <a class="txt" href="gobrd12.html#E2:12"><i><font color="Green">E41</font></i></a>, <a class="txt" href="gobrd12.html#E4:12"><i><font color="Green">E42</font></i></a>, <a class="txt" href="gobrd12.html#E7"><i><font color="Green">Lemma43</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E34:12"/><b>then </b><i><font color="Green">E83</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">Mm</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="gobrd10.html#T9">GOBRD10:9</a></i>;<br/>
<b>thus </b><a NAME="E35:12"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> holds <br/> <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>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:12_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">kx1</font> = <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>, <font color="Maroon">kx2</font> = <font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">kk1</font> = <font color="Maroon">kx1</font> <a href="nat_1.html#K1">+</a> 1, <font color="Maroon">kk2</font> = <font color="Maroon">kx2</font> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E4:12_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="gobrd12.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E5:12_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:12_2"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:12_2"/><b>then </b>
<font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="gobrd12.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E8:12_2"/><b>then </b>
( <font color="Maroon">kx1</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">kx2</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> )
 
<b>by </b><i>, <a class="txt" href="gobrd12.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="gobrd12.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E9:12_2"/><b>then </b><i><font color="Green">E86</font></i>: 
( <font color="Maroon">kk1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">kk2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:12_2"/>
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk1</font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E11:12_2"/><b>then </b><i><font color="Green">E87</font></i>: 
( <font color="Maroon">kk1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">kk2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="gobrd12.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:12_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">Mm</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">kk1</font>,<font color="Maroon">kk2</font> <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><a class="txt" href="gobrd12.html#E1"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<a NAME="E13:12_2"/><i><font color="Green">E89</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Mm</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E14:12_2"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Mm</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:12_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">Mm</font> is    <a href="matrix_1.html#M1">Matrix</a> of  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Mm</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1, <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E16:12_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Mm</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:12_2"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Mm</font></span>)</span>
 
<b>by </b><i><a class="txt" href="gobrd12.html#E2:12"><i><font color="Green">E41</font></i></a>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/>
<a NAME="E18:12_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">kk1</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">kk2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E19:12_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">kk1</font>,<font color="Maroon">kk2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">Mm</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">Mm</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="gobrd12.html#E1:12"><i><font color="Green">E37</font></i></a>, <a class="ref" href="gobrd12.html#T4" target="_self">Th4</a>, <a class="txt" href="gobrd12.html#E4:12"><i><font color="Green">E42</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E20:12_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">kk1</font>,<font color="Maroon">kk2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">Mm</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E21:12_2"/>
 <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>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></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>, , <a class="txt" href="gobrd12.html#E7"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div>
