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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2_1_1"/>
( ( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> ) or ( <font color="Maroon" title="c3">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> ) or ( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> ) or ( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E12:22"><i><font color="Green" title="E25">A7</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1"><b>suppose </b></a><a NAME="E1:22_2_1_1_1"/><i><font color="Green" title="E26">A12</font></i>: 
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> )
 ;<br/><div class="add"><b>take </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">j1</font>
;<br/><b>let </b><font color="Maroon" title="c7">i1'</font>, <font color="Maroon" title="c8">j1'</font>, <font color="Maroon" title="c9">i2'</font>, <font color="Maroon" title="c10">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="E2:22_2_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</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="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font> &amp; <font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font> )
 ;<br/><a NAME="E3:22_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1'</font> &amp; <font color="Maroon" title="c5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j1'</font> &amp; <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> )
 <b>by </b><i><a class="txt" href="goboard5.html#E6:22"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:22"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><a NAME="E4:22_2_1_1_1"/>
( ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c7">i1'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j1'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i1'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j2'</font> ) )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:22_2_1_1_1"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_2"><b>suppose </b></a><a NAME="E1:22_2_1_1_2"/><i><font color="Green" title="E26">A13</font></i>: 
( <font color="Maroon" title="c3">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> )
 ;<br/><div class="add"><b>take </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c3">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
;<br/><b>let </b><font color="Maroon" title="c7">i1'</font>, <font color="Maroon" title="c8">j1'</font>, <font color="Maroon" title="c9">i2'</font>, <font color="Maroon" title="c10">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="E2:22_2_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</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="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font> &amp; <font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font> )
 ;<br/><a NAME="E3:22_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1'</font> &amp; <font color="Maroon" title="c5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j1'</font> &amp; <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> )
 <b>by </b><i><a class="txt" href="goboard5.html#E6:22"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:22"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><a NAME="E4:22_2_1_1_2"/>
( ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c3">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c3">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c7">i1'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j1'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c3">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c3">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i1'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j2'</font> ) )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:22_2_1_1_2"><i><font color="Green" title="E26">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_3"><b>suppose </b></a><a NAME="E1:22_2_1_1_3"/><i><font color="Green" title="E26">A14</font></i>: 
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> )
 ;<br/><div class="add"><b>take </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c5">i2</font>,<font color="Maroon" title="c6">j2</font>
;<br/><b>let </b><font color="Maroon" title="c7">i1'</font>, <font color="Maroon" title="c8">j1'</font>, <font color="Maroon" title="c9">i2'</font>, <font color="Maroon" title="c10">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="E2:22_2_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</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="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font> &amp; <font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font> )
 ;<br/><a NAME="E3:22_2_1_1_3"/><b>then </b>
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1'</font> &amp; <font color="Maroon" title="c5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j1'</font> &amp; <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> )
 <b>by </b><i><a class="txt" href="goboard5.html#E6:22"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:22"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><a NAME="E4:22_2_1_1_3"/>
( ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c5">i2</font>,<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c5">i2</font>,<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c7">i1'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j1'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c5">i2</font>,<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c5">i2</font>,<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i1'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j2'</font> ) )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:22_2_1_1_3"><i><font color="Green" title="E26">A14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_4"><b>suppose </b></a><a NAME="E1:22_2_1_1_4"/><i><font color="Green" title="E26">A15</font></i>: 
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>take </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c6">j2</font>
;<br/><b>let </b><font color="Maroon" title="c7">i1'</font>, <font color="Maroon" title="c8">j1'</font>, <font color="Maroon" title="c9">i2'</font>, <font color="Maroon" title="c10">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="E2:22_2_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</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="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font> &amp; <font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font> )
 ;<br/><a NAME="E3:22_2_1_1_4"/><b>then </b>
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1'</font> &amp; <font color="Maroon" title="c5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j1'</font> &amp; <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> )
 <b>by </b><i><a class="txt" href="goboard5.html#E6:22"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:22"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><a NAME="E4:22_2_1_1_4"/>
( ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c7">i1'</font>,<font color="Maroon" title="c8">j1'</font> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c7">i1'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j1'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c9">i2'</font>,<font color="Maroon" title="c10">j2'</font> ) or ( <font color="Maroon" title="c7">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2'</font> &amp; <font color="Maroon" title="c8">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i1'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j2'</font> ) )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:22_2_1_1_4"><i><font color="Green" title="E26">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
