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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a><br/>  for <font color="Olive" title="b2">f</font> being   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b1">G</font> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">n</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">n</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> implies (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 <span class="kw">and </span><br/><a NAME="E2:40"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">i1</font>, <font color="Maroon" title="c5">j1</font>, <font color="Maroon" title="c6">i2</font>, <font color="Maroon" title="c7">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:40"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E5:40"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>)
 <span class="kw">and </span><br/><a NAME="E6:40"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E7:40"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)
 <span class="kw">and </span><br/><a NAME="E8:40"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
( ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/>
<a NAME="E9:40"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E10:40"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
( <font color="Maroon" title="c5">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">j1</font> &amp; <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E11:40"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
( <font color="Maroon" title="c4">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">i1</font> &amp; <font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">i2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E12:40"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
<font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E13:40"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
<font color="Maroon" title="c5">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E14:40"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<font color="Maroon" title="c6">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E15:40"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E16:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/>
<a NAME="E17:40"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E18:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/>
<a NAME="E19:40"/><span class="lab"><font color="Green" title="E49">A18</font></span>: 
<font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">F</font> =  <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font>;<br/>
<a NAME="E20:40"/><span class="lab"><font color="Green" title="E50">A19</font></span>: 
 <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="gobrd13.html#T20" target="_self" title="GOBRD13:th.20">Th21</a></span>;<br/>
<a NAME="E21:40"/>
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard5.html#D5" title="GOBOARD5:def.5">GOBOARD5:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">m</font>, <font color="Maroon" title="c10">k</font>, <font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c12">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E23:40"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c10">k</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E24:40"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c10">k</font>)
 <span class="kw">and </span><br/><a NAME="E25:40"/><span class="lab"><font color="Green" title="E53">A22</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E26:40"/><span class="lab"><font color="Green" title="E54">A23</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>)
 <span class="kw">and </span><br/><a NAME="E27:40"/>
( ( <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> ) or ( <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> ) or ( <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> ) or ( <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/>
<a NAME="E28:40"/><span class="lab"><font color="Green" title="E55">A24</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E29:40"/><span class="lab"><font color="Green" title="E56">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E30:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i1</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:40"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/>
<a NAME="E31:40"/><span class="lab"><font color="Green" title="E58">A27</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:40"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="txt" href="#E19:40"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/>
<a NAME="E32:40"/><span class="lab"><font color="Green" title="E59">A28</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E33:40"/><span class="lab"><font color="Green" title="E60">A29</font></span>: 
<font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E34:40"/><span class="lab"><font color="Green" title="E61">A30</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E35:40"/><span class="lab"><font color="Green" title="E62">A31</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E36:40"/><span class="lab"><font color="Green" title="E63">A32</font></span>: 
<font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E37:40"/><span class="lab"><font color="Green" title="E64">A33</font></span>: 
<font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:40"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E38:40"/><span class="lab"><font color="Green" title="E65">A34</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E39:40"/><span class="lab"><font color="Green" title="E66">A35</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E40:40"/><span class="lab"><font color="Green" title="E67">A36</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:40"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E41:40"/><span class="lab"><font color="Green" title="E68">A37</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E42:40"/><span class="lab"><font color="Green" title="E69">A38</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E43:40"/><span class="lab"><font color="Green" title="E70">A39</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E44:40"/><span class="lab"><font color="Green" title="E71">A40</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:40"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E45:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A41</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c12">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c12">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/>
<a NAME="E46:40"/><span class="lab"><font color="Green" title="E73">A42</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:40"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E47:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A43</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="txt" href="#E44:40"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/>
<a NAME="E48:40"/><span class="lab"><font color="Green" title="E75">A44</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c12">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:40"><span class="lab"><font color="Green" title="E73">A42</font></span></a>, <a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="txt" href="#E44:40"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/>
<a NAME="E49:40"/><span class="lab"><font color="Green" title="E76">A45</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E50:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A46</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">k</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E46:40"><span class="lab"><font color="Green" title="E73">A42</font></span></a>, <a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1"/>
( ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E38">A7</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1"/><span class="lab"><font color="Green" title="E78">A47</font></span>: 
( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E79">A48</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:40_1_1_1"/><span class="lab"><font color="Green" title="E80">A49</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:40_1_1_1"/><span class="lab"><font color="Green" title="E81">A50</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:40_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A51</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">l</font>, <font color="Maroon" title="c14">i9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:40_1_1_1"/><span class="lab"><font color="Green" title="E83">A52</font></span>: 
<font color="Maroon" title="c13">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E6:40_1_1_1"/><span class="lab"><font color="Green" title="E84">A53</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:40_1_1_1"/><span class="lab"><font color="Green" title="E85">A54</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">i9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T8" title="JORDAN5D:th.8">JORDAN5D:8</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E8:40_1_1_1"/><span class="lab"><font color="Green" title="E86">A55</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E35:40"><span class="lab"><font color="Green" title="E62">A31</font></span></a>, <a class="txt" href="#E3:40_1_1_1"><span class="lab"><font color="Green" title="E82">A51</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><a NAME="E9:40_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i9</font> &amp; <font color="Maroon" title="c14">i9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_1_1"><span class="lab"><font color="Green" title="E84">A53</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E10:40_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A56</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">i9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:40"><span class="lab"><font color="Green" title="E62">A31</font></span></a>, <a class="txt" href="#E3:40_1_1_1"><span class="lab"><font color="Green" title="E82">A51</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">i19</font>, <font color="Maroon" title="c16">j19</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E12:40_1_1_1"/><span class="lab"><font color="Green" title="E88">A57</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</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> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E13:40_1_1_1"/><span class="lab"><font color="Green" title="E89">A58</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E5:40_1_1_1"><span class="lab"><font color="Green" title="E83">A52</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/><a NAME="E14:40_1_1_1"/><span class="lab"><font color="Green" title="E90">A59</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_1_1_1"><span class="lab"><font color="Green" title="E88">A57</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E15:40_1_1_1"/><span class="lab"><font color="Green" title="E91">A60</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i19</font> &amp; <font color="Maroon" title="c15">i19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_1_1_1"><span class="lab"><font color="Green" title="E88">A57</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E16:40_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A61</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><div><span class="lab"><font color="Green" title="E93">A62</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_1_1_1"/>
<font color="Maroon" title="c5">j1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c16">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E31:40"><span class="lab"><font color="Green" title="E58">A27</font></span></a>, <a class="txt" href="#E15:40_1_1_1"><span class="lab"><font color="Green" title="E91">A60</font></span></a>, <a class="txt" href="#E14:40_1_1_1"><span class="lab"><font color="Green" title="E90">A59</font></span></a>, <a class="txt" href="#E16:40_1_1_1"><span class="lab"><font color="Green" title="E92">A61</font></span></a>, <a class="ref" href="sprect_3.html#T12" title="SPRECT_3:th.12">SPRECT_3:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E28:40"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E49:40"><span class="lab"><font color="Green" title="E76">A45</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E34:40"><span class="lab"><font color="Green" title="E61">A30</font></span></a>, <a class="txt" href="#E3:40_1_1_1"><span class="lab"><font color="Green" title="E82">A51</font></span></a>, <a class="txt" href="#E7:40_1_1_1"><span class="lab"><font color="Green" title="E85">A54</font></span></a>, <a class="txt" href="#E10:40_1_1_1"><span class="lab"><font color="Green" title="E87">A56</font></span></a>, <a class="txt" href="#E8:40_1_1_1"><span class="lab"><font color="Green" title="E86">A55</font></span></a>, <a class="txt" href="#E13:40_1_1_1"><span class="lab"><font color="Green" title="E89">A58</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E18:40_1_1_1"/><span class="lab"><font color="Green" title="E94">A63</font></span>: 
<font color="Maroon" title="c16">j19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_1_1_1"><span class="lab"><font color="Green" title="E88">A57</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E19:40_1_1_1"/><span class="lab"><font color="Green" title="E95">A64</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E15:40_1_1_1"><span class="lab"><font color="Green" title="E91">A60</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_1_1_2"/>
<font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E15:40_1_1_1"><span class="lab"><font color="Green" title="E91">A60</font></span></a>, <a class="txt" href="#E18:40_1_1_1"><span class="lab"><font color="Green" title="E94">A63</font></span></a>, <a class="txt" href="#E1:40_1_1_1"><span class="lab"><font color="Green" title="E80">A49</font></span></a>, <a class="txt" href="#E19:40_1_1_1"><span class="lab"><font color="Green" title="E95">A64</font></span></a>, <a class="ref" href="sprect_3.html#T12" title="SPRECT_3:th.12">SPRECT_3:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E28:40"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="txt" href="#E48:40"><span class="lab"><font color="Green" title="E75">A44</font></span></a>, <a class="txt" href="#E45:40"><span class="lab"><font color="Green" title="E72">A41</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E35:40"><span class="lab"><font color="Green" title="E62">A31</font></span></a>, <a class="txt" href="#E2:40_1_1_1"><span class="lab"><font color="Green" title="E81">A50</font></span></a>, <a class="txt" href="#E7:40_1_1_1"><span class="lab"><font color="Green" title="E85">A54</font></span></a>, <a class="txt" href="#E10:40_1_1_1"><span class="lab"><font color="Green" title="E87">A56</font></span></a>, <a class="txt" href="#E8:40_1_1_1"><span class="lab"><font color="Green" title="E86">A55</font></span></a>, <a class="txt" href="#E13:40_1_1_1"><span class="lab"><font color="Green" title="E89">A58</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E21:40_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1"><span class="lab"><font color="Green" title="E78">A47</font></span></a>, <a class="txt" href="#E17:40_1_1_1"><span class="lab"><font color="Green" title="E93">A62</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A65</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c10">k</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E44:40"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="txt" href="#E48:40"><span class="lab"><font color="Green" title="E75">A44</font></span></a>, <a class="txt" href="#E45:40"><span class="lab"><font color="Green" title="E72">A41</font></span></a>, <a class="ref" href="sprect_3.html#T12" title="SPRECT_3:th.12">SPRECT_3:12</a></span>;<br/><a NAME="E3:40_1_1_2"/>
<font color="Maroon" title="c5">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1"><span class="lab"><font color="Green" title="E78">A47</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E31:40"><span class="lab"><font color="Green" title="E58">A27</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E2:40_1_1_2"><span class="lab"><font color="Green" title="E80">A65</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:40_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:40_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1"><span class="lab"><font color="Green" title="E79">A48</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A66</font></span>: 
(  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c10">k</font>) &amp;  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">k</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E25:40"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E34:40"><span class="lab"><font color="Green" title="E61">A30</font></span></a>, <a class="txt" href="#E33:40"><span class="lab"><font color="Green" title="E60">A29</font></span></a>, <a class="ref" href="goboard5.html#D6" title="GOBOARD5:def.6">GOBOARD5:def 6</a>, <a class="ref" href="goboard5.html#D7" title="GOBOARD5:def.7">GOBOARD5:def 7</a></span>;<br/><a NAME="E7:40_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>) &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c5">j1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E1:40_1_1"><span class="lab"><font color="Green" title="E78">A47</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:40_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:40"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E6:40_1_1"><span class="lab"><font color="Green" title="E80">A66</font></span></a>, <a class="ref" href="gobrd13.html#T17" target="_self" title="GOBRD13:th.17">Th18</a>, <a class="ref" href="gobrd13.html#T18" target="_self" title="GOBRD13:th.18">Th19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_2"/><span class="lab"><font color="Green" title="E78">A67</font></span>: 
( <font color="Maroon" title="c4">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E79">A68</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_2_1"/><span class="lab"><font color="Green" title="E80">A69</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A70</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">l</font>, <font color="Maroon" title="c14">j9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:40_1_2_1"/><span class="lab"><font color="Green" title="E82">A71</font></span>: 
<font color="Maroon" title="c13">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E5:40_1_2_1"/><span class="lab"><font color="Green" title="E83">A72</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j9</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:40_1_2_1"/><span class="lab"><font color="Green" title="E84">A73</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j9</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T7" title="JORDAN5D:th.7">JORDAN5D:7</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E7:40_1_2_1"/><span class="lab"><font color="Green" title="E85">A74</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">k</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:40"><span class="lab"><font color="Green" title="E76">A45</font></span></a>, <a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E42:40"><span class="lab"><font color="Green" title="E69">A38</font></span></a>, <a class="txt" href="#E2:40_1_2_1"><span class="lab"><font color="Green" title="E81">A70</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E8:40_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j9</font> &amp; <font color="Maroon" title="c14">j9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_2_1"><span class="lab"><font color="Green" title="E83">A72</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E9:40_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A75</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j9</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:40"><span class="lab"><font color="Green" title="E69">A38</font></span></a>, <a class="txt" href="#E2:40_1_2_1"><span class="lab"><font color="Green" title="E81">A70</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E10:40_1_2_1"/><span class="lab"><font color="Green" title="E87">A76</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:40"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="txt" href="#E42:40"><span class="lab"><font color="Green" title="E69">A38</font></span></a>, <a class="txt" href="#E2:40_1_2_1"><span class="lab"><font color="Green" title="E81">A70</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E11:40_1_2_1"/><span class="lab"><font color="Green" title="E88">A77</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">i19</font>, <font color="Maroon" title="c16">j19</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E13:40_1_2_1"/><span class="lab"><font color="Green" title="E89">A78</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</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> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E14:40_1_2_1"/><span class="lab"><font color="Green" title="E90">A79</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E4:40_1_2_1"><span class="lab"><font color="Green" title="E82">A71</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/><a NAME="E15:40_1_2_1"/><span class="lab"><font color="Green" title="E91">A80</font></span>: 
<font color="Maroon" title="c15">i19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_1_2_1"><span class="lab"><font color="Green" title="E89">A78</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E16:40_1_2_1"/><span class="lab"><font color="Green" title="E92">A81</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j19</font> &amp; <font color="Maroon" title="c16">j19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_1_2_1"><span class="lab"><font color="Green" title="E89">A78</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E17:40_1_2_1"/><span class="lab"><font color="Green" title="E93">A82</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_1_2_1"><span class="lab"><font color="Green" title="E89">A78</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E18:40_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A83</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40_1_2_1"><span class="lab"><font color="Green" title="E91">A80</font></span></a>, <a class="txt" href="#E16:40_1_2_1"><span class="lab"><font color="Green" title="E92">A81</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E19:40_1_2_1"/><span class="lab"><font color="Green" title="E95">A84</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E17:40_1_2_1"><span class="lab"><font color="Green" title="E93">A82</font></span></a>, <a class="txt" href="#E15:40_1_2_1"><span class="lab"><font color="Green" title="E91">A80</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><div><span class="lab"><font color="Green" title="E96">A85</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_2_1_1"/>
<font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:40"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E17:40_1_2_1"><span class="lab"><font color="Green" title="E93">A82</font></span></a>, <a class="txt" href="#E18:40_1_2_1"><span class="lab"><font color="Green" title="E94">A83</font></span></a>, <a class="txt" href="#E19:40_1_2_1"><span class="lab"><font color="Green" title="E95">A84</font></span></a>, <a class="ref" href="sprect_3.html#T13" title="SPRECT_3:th.13">SPRECT_3:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E28:40"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E49:40"><span class="lab"><font color="Green" title="E76">A45</font></span></a>, <a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E39:40"><span class="lab"><font color="Green" title="E66">A35</font></span></a>, <a class="txt" href="#E2:40_1_2_1"><span class="lab"><font color="Green" title="E81">A70</font></span></a>, <a class="txt" href="#E6:40_1_2_1"><span class="lab"><font color="Green" title="E84">A73</font></span></a>, <a class="txt" href="#E9:40_1_2_1"><span class="lab"><font color="Green" title="E86">A75</font></span></a>, <a class="txt" href="#E7:40_1_2_1"><span class="lab"><font color="Green" title="E85">A74</font></span></a>, <a class="txt" href="#E14:40_1_2_1"><span class="lab"><font color="Green" title="E90">A79</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E21:40_1_2_1"/><span class="lab"><font color="Green" title="E97">A86</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E16:40_1_2_1"><span class="lab"><font color="Green" title="E92">A81</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_2_1_2"/>
<font color="Maroon" title="c6">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E15:40_1_2_1"><span class="lab"><font color="Green" title="E91">A80</font></span></a>, <a class="txt" href="#E16:40_1_2_1"><span class="lab"><font color="Green" title="E92">A81</font></span></a>, <a class="txt" href="#E11:40_1_2_1"><span class="lab"><font color="Green" title="E88">A77</font></span></a>, <a class="txt" href="#E21:40_1_2_1"><span class="lab"><font color="Green" title="E97">A86</font></span></a>, <a class="ref" href="sprect_3.html#T13" title="SPRECT_3:th.13">SPRECT_3:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="txt" href="#E44:40"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E1:40_1_2_1"><span class="lab"><font color="Green" title="E80">A69</font></span></a>, <a class="txt" href="#E6:40_1_2_1"><span class="lab"><font color="Green" title="E84">A73</font></span></a>, <a class="txt" href="#E9:40_1_2_1"><span class="lab"><font color="Green" title="E86">A75</font></span></a>, <a class="txt" href="#E10:40_1_2_1"><span class="lab"><font color="Green" title="E87">A76</font></span></a>, <a class="txt" href="#E14:40_1_2_1"><span class="lab"><font color="Green" title="E90">A79</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E23:40_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_2"><span class="lab"><font color="Green" title="E78">A67</font></span></a>, <a class="txt" href="#E20:40_1_2_1"><span class="lab"><font color="Green" title="E96">A85</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_2_2"/>
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A87</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c10">k</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E49:40"><span class="lab"><font color="Green" title="E76">A45</font></span></a>, <a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E46:40"><span class="lab"><font color="Green" title="E73">A42</font></span></a>, <a class="txt" href="#E47:40"><span class="lab"><font color="Green" title="E74">A43</font></span></a>, <a class="txt" href="#E50:40"><span class="lab"><font color="Green" title="E77">A46</font></span></a>, <a class="ref" href="sprect_3.html#T13" title="SPRECT_3:th.13">SPRECT_3:13</a></span>;<br/><a NAME="E3:40_1_2_2"/>
<font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_2"><span class="lab"><font color="Green" title="E78">A67</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E29:40"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E1:40_1_2"><span class="lab"><font color="Green" title="E78">A67</font></span></a>, <a class="txt" href="#E2:40_1_2_2"><span class="lab"><font color="Green" title="E80">A87</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:40_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:40_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_2"><span class="lab"><font color="Green" title="E79">A68</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:40_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A88</font></span>: 
(  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) &amp;  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c10">k</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E25:40"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E39:40"><span class="lab"><font color="Green" title="E66">A35</font></span></a>, <a class="txt" href="#E38:40"><span class="lab"><font color="Green" title="E65">A34</font></span></a>, <a class="ref" href="goboard5.html#D6" title="GOBOARD5:def.6">GOBOARD5:def 6</a>, <a class="ref" href="goboard5.html#D7" title="GOBOARD5:def.7">GOBOARD5:def 7</a></span>;<br/><a NAME="E7:40_1_2"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E1:40_1_2"><span class="lab"><font color="Green" title="E78">A67</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:40_1_2"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:40"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E6:40_1_2"><span class="lab"><font color="Green" title="E80">A88</font></span></a>, <a class="ref" href="gobrd13.html#T17" target="_self" title="GOBRD13:th.17">Th18</a>, <a class="ref" href="gobrd13.html#T19" target="_self" title="GOBRD13:th.19">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_3"><span class="kw">suppose </span></a><a NAME="E1:40_1_3"/><span class="lab"><font color="Green" title="E78">A89</font></span>: 
( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E79">A90</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_3_1"/><span class="lab"><font color="Green" title="E80">A91</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A92</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">l</font>, <font color="Maroon" title="c14">j9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:40_1_3_1"/><span class="lab"><font color="Green" title="E82">A93</font></span>: 
<font color="Maroon" title="c13">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E5:40_1_3_1"/><span class="lab"><font color="Green" title="E83">A94</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j9</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:40_1_3_1"/><span class="lab"><font color="Green" title="E84">A95</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j9</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T7" title="JORDAN5D:th.7">JORDAN5D:7</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E7:40_1_3_1"/><span class="lab"><font color="Green" title="E85">A96</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">k</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:40"><span class="lab"><font color="Green" title="E76">A45</font></span></a>, <a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E41:40"><span class="lab"><font color="Green" title="E68">A37</font></span></a>, <a class="txt" href="#E2:40_1_3_1"><span class="lab"><font color="Green" title="E81">A92</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E8:40_1_3_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j9</font> &amp; <font color="Maroon" title="c14">j9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_3_1"><span class="lab"><font color="Green" title="E83">A94</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E9:40_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A97</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j9</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:40"><span class="lab"><font color="Green" title="E68">A37</font></span></a>, <a class="txt" href="#E2:40_1_3_1"><span class="lab"><font color="Green" title="E81">A92</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E10:40_1_3_1"/><span class="lab"><font color="Green" title="E87">A98</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:40"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="txt" href="#E41:40"><span class="lab"><font color="Green" title="E68">A37</font></span></a>, <a class="txt" href="#E2:40_1_3_1"><span class="lab"><font color="Green" title="E81">A92</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E11:40_1_3_1"/><span class="lab"><font color="Green" title="E88">A99</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">i19</font>, <font color="Maroon" title="c16">j19</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E13:40_1_3_1"/><span class="lab"><font color="Green" title="E89">A100</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</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> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E14:40_1_3_1"/><span class="lab"><font color="Green" title="E90">A101</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E4:40_1_3_1"><span class="lab"><font color="Green" title="E82">A93</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/><a NAME="E15:40_1_3_1"/><span class="lab"><font color="Green" title="E91">A102</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_1_3_1"><span class="lab"><font color="Green" title="E89">A100</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E16:40_1_3_1"/><span class="lab"><font color="Green" title="E92">A103</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j19</font> &amp; <font color="Maroon" title="c16">j19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_1_3_1"><span class="lab"><font color="Green" title="E89">A100</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E17:40_1_3_1"/><span class="lab"><font color="Green" title="E93">A104</font></span>: 
<font color="Maroon" title="c15">i19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_1_3_1"><span class="lab"><font color="Green" title="E89">A100</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E18:40_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A105</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40_1_3_1"><span class="lab"><font color="Green" title="E91">A102</font></span></a>, <a class="txt" href="#E16:40_1_3_1"><span class="lab"><font color="Green" title="E92">A103</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E19:40_1_3_1"/><span class="lab"><font color="Green" title="E95">A106</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E15:40_1_3_1"><span class="lab"><font color="Green" title="E91">A102</font></span></a>, <a class="txt" href="#E17:40_1_3_1"><span class="lab"><font color="Green" title="E93">A104</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><div><span class="lab"><font color="Green" title="E96">A107</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_3_1_1"/>
<font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:40"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E17:40_1_3_1"><span class="lab"><font color="Green" title="E93">A104</font></span></a>, <a class="txt" href="#E18:40_1_3_1"><span class="lab"><font color="Green" title="E94">A105</font></span></a>, <a class="txt" href="#E19:40_1_3_1"><span class="lab"><font color="Green" title="E95">A106</font></span></a>, <a class="ref" href="sprect_3.html#T13" title="SPRECT_3:th.13">SPRECT_3:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E49:40"><span class="lab"><font color="Green" title="E76">A45</font></span></a>, <a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E1:40_1_3_1"><span class="lab"><font color="Green" title="E80">A91</font></span></a>, <a class="txt" href="#E6:40_1_3_1"><span class="lab"><font color="Green" title="E84">A95</font></span></a>, <a class="txt" href="#E9:40_1_3_1"><span class="lab"><font color="Green" title="E86">A97</font></span></a>, <a class="txt" href="#E7:40_1_3_1"><span class="lab"><font color="Green" title="E85">A96</font></span></a>, <a class="txt" href="#E14:40_1_3_1"><span class="lab"><font color="Green" title="E90">A101</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E21:40_1_3_1"/><span class="lab"><font color="Green" title="E97">A108</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,1)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E16:40_1_3_1"><span class="lab"><font color="Green" title="E92">A103</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_3_1_2"/>
<font color="Maroon" title="c6">i2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E15:40_1_3_1"><span class="lab"><font color="Green" title="E91">A102</font></span></a>, <a class="txt" href="#E16:40_1_3_1"><span class="lab"><font color="Green" title="E92">A103</font></span></a>, <a class="txt" href="#E11:40_1_3_1"><span class="lab"><font color="Green" title="E88">A99</font></span></a>, <a class="txt" href="#E21:40_1_3_1"><span class="lab"><font color="Green" title="E97">A108</font></span></a>, <a class="ref" href="sprect_3.html#T13" title="SPRECT_3:th.13">SPRECT_3:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E46:40"><span class="lab"><font color="Green" title="E73">A42</font></span></a>, <a class="txt" href="#E44:40"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E38:40"><span class="lab"><font color="Green" title="E65">A34</font></span></a>, <a class="txt" href="#E2:40_1_3_1"><span class="lab"><font color="Green" title="E81">A92</font></span></a>, <a class="txt" href="#E6:40_1_3_1"><span class="lab"><font color="Green" title="E84">A95</font></span></a>, <a class="txt" href="#E9:40_1_3_1"><span class="lab"><font color="Green" title="E86">A97</font></span></a>, <a class="txt" href="#E10:40_1_3_1"><span class="lab"><font color="Green" title="E87">A98</font></span></a>, <a class="txt" href="#E14:40_1_3_1"><span class="lab"><font color="Green" title="E90">A101</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E23:40_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_3"><span class="lab"><font color="Green" title="E78">A89</font></span></a>, <a class="txt" href="#E20:40_1_3_1"><span class="lab"><font color="Green" title="E96">A107</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_3_2"/>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A109</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c10">k</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E49:40"><span class="lab"><font color="Green" title="E76">A45</font></span></a>, <a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="txt" href="#E47:40"><span class="lab"><font color="Green" title="E74">A43</font></span></a>, <a class="txt" href="#E50:40"><span class="lab"><font color="Green" title="E77">A46</font></span></a>, <a class="ref" href="sprect_3.html#T13" title="SPRECT_3:th.13">SPRECT_3:13</a></span>;<br/><a NAME="E3:40_1_3_2"/>
<font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_3"><span class="lab"><font color="Green" title="E78">A89</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_1_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E19:40"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E1:40_1_3"><span class="lab"><font color="Green" title="E78">A89</font></span></a>, <a class="txt" href="#E2:40_1_3_2"><span class="lab"><font color="Green" title="E80">A109</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:40_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:40_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_3"><span class="lab"><font color="Green" title="E79">A90</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A110</font></span>: 
(  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>) &amp;  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E25:40"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E39:40"><span class="lab"><font color="Green" title="E66">A35</font></span></a>, <a class="txt" href="#E38:40"><span class="lab"><font color="Green" title="E65">A34</font></span></a>, <a class="ref" href="goboard5.html#D6" title="GOBOARD5:def.6">GOBOARD5:def 6</a>, <a class="ref" href="goboard5.html#D7" title="GOBOARD5:def.7">GOBOARD5:def 7</a></span>;<br/><a NAME="E7:40_1_3"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>) &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c6">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E1:40_1_3"><span class="lab"><font color="Green" title="E78">A89</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:40_1_3"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:40"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E25:40"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E6:40_1_3"><span class="lab"><font color="Green" title="E80">A110</font></span></a>, <a class="ref" href="gobrd13.html#T17" target="_self" title="GOBRD13:th.17">Th18</a>, <a class="ref" href="gobrd13.html#T19" target="_self" title="GOBRD13:th.19">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_4"><span class="kw">suppose </span></a><a NAME="E1:40_1_4"/><span class="lab"><font color="Green" title="E78">A111</font></span>: 
( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E79">A112</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:40_1_4_1"/><span class="lab"><font color="Green" title="E80">A113</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:40_1_4_1"/><span class="lab"><font color="Green" title="E81">A114</font></span>: 
<font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:40_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A115</font></span>: 
<font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">l</font>, <font color="Maroon" title="c14">i9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:40_1_4_1"/><span class="lab"><font color="Green" title="E83">A116</font></span>: 
<font color="Maroon" title="c13">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E6:40_1_4_1"/><span class="lab"><font color="Green" title="E84">A117</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:40_1_4_1"/><span class="lab"><font color="Green" title="E85">A118</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">i9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T8" title="JORDAN5D:th.8">JORDAN5D:8</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E8:40_1_4_1"/><span class="lab"><font color="Green" title="E86">A119</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E36:40"><span class="lab"><font color="Green" title="E63">A32</font></span></a>, <a class="txt" href="#E3:40_1_4_1"><span class="lab"><font color="Green" title="E82">A115</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><a NAME="E9:40_1_4_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i9</font> &amp; <font color="Maroon" title="c14">i9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_4_1"><span class="lab"><font color="Green" title="E84">A117</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E10:40_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A120</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">i9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:40"><span class="lab"><font color="Green" title="E63">A32</font></span></a>, <a class="txt" href="#E3:40_1_4_1"><span class="lab"><font color="Green" title="E82">A115</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">i19</font>, <font color="Maroon" title="c16">j19</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E12:40_1_4_1"/><span class="lab"><font color="Green" title="E88">A121</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</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> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E13:40_1_4_1"/><span class="lab"><font color="Green" title="E89">A122</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E5:40_1_4_1"><span class="lab"><font color="Green" title="E83">A116</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/><a NAME="E14:40_1_4_1"/><span class="lab"><font color="Green" title="E90">A123</font></span>: 
<font color="Maroon" title="c16">j19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_1_4_1"><span class="lab"><font color="Green" title="E88">A121</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E15:40_1_4_1"/><span class="lab"><font color="Green" title="E91">A124</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i19</font> &amp; <font color="Maroon" title="c15">i19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_1_4_1"><span class="lab"><font color="Green" title="E88">A121</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E16:40_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A125</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><div><span class="lab"><font color="Green" title="E93">A126</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_4_1_1"/>
<font color="Maroon" title="c5">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_4_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E31:40"><span class="lab"><font color="Green" title="E58">A27</font></span></a>, <a class="txt" href="#E15:40_1_4_1"><span class="lab"><font color="Green" title="E91">A124</font></span></a>, <a class="txt" href="#E14:40_1_4_1"><span class="lab"><font color="Green" title="E90">A123</font></span></a>, <a class="txt" href="#E16:40_1_4_1"><span class="lab"><font color="Green" title="E92">A125</font></span></a>, <a class="ref" href="sprect_3.html#T12" title="SPRECT_3:th.12">SPRECT_3:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E28:40"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E36:40"><span class="lab"><font color="Green" title="E63">A32</font></span></a>, <a class="txt" href="#E2:40_1_4_1"><span class="lab"><font color="Green" title="E81">A114</font></span></a>, <a class="txt" href="#E7:40_1_4_1"><span class="lab"><font color="Green" title="E85">A118</font></span></a>, <a class="txt" href="#E10:40_1_4_1"><span class="lab"><font color="Green" title="E87">A120</font></span></a>, <a class="txt" href="#E8:40_1_4_1"><span class="lab"><font color="Green" title="E86">A119</font></span></a>, <a class="txt" href="#E13:40_1_4_1"><span class="lab"><font color="Green" title="E89">A122</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E18:40_1_4_1"/><span class="lab"><font color="Green" title="E94">A127</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:40"><span class="lab"><font color="Green" title="E73">A42</font></span></a>, <a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="txt" href="#E36:40"><span class="lab"><font color="Green" title="E63">A32</font></span></a>, <a class="txt" href="#E3:40_1_4_1"><span class="lab"><font color="Green" title="E82">A115</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><a NAME="E19:40_1_4_1"/><span class="lab"><font color="Green" title="E95">A128</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_1_4_1"><span class="lab"><font color="Green" title="E88">A121</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E20:40_1_4_1"/><span class="lab"><font color="Green" title="E96">A129</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E15:40_1_4_1"><span class="lab"><font color="Green" title="E91">A124</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_4_1_2"/>
<font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c16">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c15">i19</font>,<font color="Maroon" title="c16">j19</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E15:40_1_4_1"><span class="lab"><font color="Green" title="E91">A124</font></span></a>, <a class="txt" href="#E19:40_1_4_1"><span class="lab"><font color="Green" title="E95">A128</font></span></a>, <a class="txt" href="#E1:40_1_4_1"><span class="lab"><font color="Green" title="E80">A113</font></span></a>, <a class="txt" href="#E20:40_1_4_1"><span class="lab"><font color="Green" title="E96">A129</font></span></a>, <a class="ref" href="sprect_3.html#T12" title="SPRECT_3:th.12">SPRECT_3:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E46:40"><span class="lab"><font color="Green" title="E73">A42</font></span></a>, <a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="txt" href="#E44:40"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E33:40"><span class="lab"><font color="Green" title="E60">A29</font></span></a>, <a class="txt" href="#E3:40_1_4_1"><span class="lab"><font color="Green" title="E82">A115</font></span></a>, <a class="txt" href="#E7:40_1_4_1"><span class="lab"><font color="Green" title="E85">A118</font></span></a>, <a class="txt" href="#E10:40_1_4_1"><span class="lab"><font color="Green" title="E87">A120</font></span></a>, <a class="txt" href="#E18:40_1_4_1"><span class="lab"><font color="Green" title="E94">A127</font></span></a>, <a class="txt" href="#E13:40_1_4_1"><span class="lab"><font color="Green" title="E89">A122</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E22:40_1_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_4"><span class="lab"><font color="Green" title="E78">A111</font></span></a>, <a class="txt" href="#E17:40_1_4_1"><span class="lab"><font color="Green" title="E93">A126</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_4_2"/>
<font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A130</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c10">k</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E49:40"><span class="lab"><font color="Green" title="E76">A45</font></span></a>, <a class="txt" href="#E37:40"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="txt" href="#E48:40"><span class="lab"><font color="Green" title="E75">A44</font></span></a>, <a class="txt" href="#E45:40"><span class="lab"><font color="Green" title="E72">A41</font></span></a>, <a class="ref" href="sprect_3.html#T12" title="SPRECT_3:th.12">SPRECT_3:12</a></span>;<br/><a NAME="E3:40_1_4_2"/>
<font color="Maroon" title="c5">j1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_4"><span class="lab"><font color="Green" title="E78">A111</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_1_4_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E17:40"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E31:40"><span class="lab"><font color="Green" title="E58">A27</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E2:40_1_4_2"><span class="lab"><font color="Green" title="E80">A130</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:40_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:40_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_4"><span class="lab"><font color="Green" title="E79">A112</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:40_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A131</font></span>: 
(  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j</font>) &amp;  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</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#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c12">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E25:40"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E34:40"><span class="lab"><font color="Green" title="E61">A30</font></span></a>, <a class="txt" href="#E33:40"><span class="lab"><font color="Green" title="E60">A29</font></span></a>, <a class="ref" href="goboard5.html#D6" title="GOBOARD5:def.6">GOBOARD5:def 6</a>, <a class="ref" href="goboard5.html#D7" title="GOBOARD5:def.7">GOBOARD5:def 7</a></span>;<br/><div><span class="lab"><font color="Green" title="E81">A132</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_4_3"/><span class="lab"><font color="Green" title="E82">A133</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_4_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1_4_3_1"/>
( <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font> or <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_4_3"><span class="lab"><font color="Green" title="E82">A133</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_4_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_4_3_1_1"/>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_4_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E28:40"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E49:40"><span class="lab"><font color="Green" title="E76">A45</font></span></a>, <a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E40:40"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="txt" href="#E47:40"><span class="lab"><font color="Green" title="E74">A43</font></span></a>, <a class="txt" href="#E50:40"><span class="lab"><font color="Green" title="E77">A46</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E30:40"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="txt" href="#E18:40"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E1:40_1_4"><span class="lab"><font color="Green" title="E78">A111</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_4_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_4_3_1_2"/>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_4_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E32:40"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E49:40"><span class="lab"><font color="Green" title="E76">A45</font></span></a>, <a class="txt" href="#E43:40"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="txt" href="#E46:40"><span class="lab"><font color="Green" title="E73">A42</font></span></a>, <a class="txt" href="#E47:40"><span class="lab"><font color="Green" title="E74">A43</font></span></a>, <a class="txt" href="#E50:40"><span class="lab"><font color="Green" title="E77">A46</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E30:40"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="txt" href="#E18:40"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E1:40_1_4"><span class="lab"><font color="Green" title="E78">A111</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E8:40_1_4"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c7">j2</font>) &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c7">j2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E1:40_1_4"><span class="lab"><font color="Green" title="E78">A111</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E9:40_1_4"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:40"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E25:40"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E26:40"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E1:40_1_4"><span class="lab"><font color="Green" title="E78">A111</font></span></a>, <a class="txt" href="#E7:40_1_4"><span class="lab"><font color="Green" title="E81">A132</font></span></a>, <a class="txt" href="#E6:40_1_4"><span class="lab"><font color="Green" title="E80">A131</font></span></a>, <a class="ref" href="gobrd13.html#T17" target="_self" title="GOBRD13:th.17">Th18</a>, <a class="ref" href="gobrd13.html#T18" target="_self" title="GOBRD13:th.18">Th19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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