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<span class="kw">let </span><font color="Maroon" title="c1">i1</font>, <font color="Maroon" title="c2">j1</font>, <font color="Maroon" title="c3">i2</font>, <font color="Maroon" title="c4">j2</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">G1</font>, <font color="Olive" title="b2">G2</font> being   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>  st  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Olive" title="b1">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Olive" title="b2">G2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">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="Olive" title="b1">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">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="Olive" title="b2">G2</font> &amp; <font color="Olive" title="b1">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">j2</font>) holds <br/> <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Olive" title="b2">G2</font>,<font color="Maroon" title="c3">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Olive" title="b1">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G2</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"> (  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c5">G1</font> <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="c6">G2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">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="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">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="c6">G2</font> &amp; <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">j2</font>) implies  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c3">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c5">G1</font> <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="c6">G2</font>
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">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="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E3:23"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">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="c6">G2</font>
 <span class="kw">and </span><br/><a NAME="E4:23"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">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="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c3">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/>

<a NAME="E5:23"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E6:23"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E21">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="E7:23"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E21">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="E8:23"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c2">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="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E9:23"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<font color="Maroon" title="c4">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="c6">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E21">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="E10:23"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c3">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="c6">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E21">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="E11:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">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="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">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="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/>
<a NAME="E12:23"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<font color="Maroon" title="c1">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="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E13:23"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E14:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c2">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="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c3">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) or <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E15:23"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c3">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/>

<a NAME="E16:23"/><span class="lab"><font color="Green" title="E34">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">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="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c4">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="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_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="E1:23_1"/>
( ( <font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j2</font> ) or ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j1</font> &amp; <font color="Maroon" title="c4">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j1</font> &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E24">A6</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="23_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1"/><span class="lab"><font color="Green" title="E35">A17</font></span>: 
( <font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A18</font></span>: 
<font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>;<br/><a NAME="E3:23_1_1"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
<font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_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="E1:23_1_1_1_1"/>
( <font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G2</font> or <font color="Maroon" title="c3">i2</font> <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> <font color="Maroon" title="c6">G2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E28">A10</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="23_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A20</font></span>: 
<font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G2</font></span>)</span>,1)</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:23"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="gobrd11.html#T27" title="GOBRD11:th.27">GOBRD11:27</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r9</font>, <font color="Maroon" title="c9">s9</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:23_1_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A21</font></span>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c8">r9</font>,<font color="Maroon" title="c9">s9</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G2</font></span>)</span>,1)</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> <font color="Maroon" title="c8">r9</font> )
 <span class="kw">and </span><br/><a NAME="E5:23_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A22</font></span>: 
<font color="Maroon" title="c9">s9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><a NAME="E6:23_1_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A23</font></span>: 
<font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:23_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="ref" href="gobrd13.html#T10" target="_self" title="GOBRD13:th.10">Th11</a></span>;<br/><a NAME="E7:23_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</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="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">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="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><a NAME="E8:23_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">s9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</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="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E5:23_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><a NAME="E9:23_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>,1)</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E1:23_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E4:23_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="txt" href="#E6:23_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A23</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:23_1_1_1_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</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="#E2:23_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E6:23_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="ref" href="gobrd11.html#T27" title="GOBRD11:th.27">GOBRD11:27</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="23_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A24</font></span>: 
<font color="Maroon" title="c3">i2</font> <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> <font color="Maroon" title="c6">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,1)</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</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>  &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:23"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="gobrd11.html#T30" title="GOBRD11:th.30">GOBRD11:30</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r9</font>, <font color="Maroon" title="c9">s9</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:23_1_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A25</font></span>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c8">r9</font>,<font color="Maroon" title="c9">s9</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,1)</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> <font color="Maroon" title="c8">r9</font> )
 <span class="kw">and </span><br/><a NAME="E5:23_1_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A26</font></span>: 
<font color="Maroon" title="c8">r9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</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">and </span><br/><a NAME="E6:23_1_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A27</font></span>: 
<font color="Maroon" title="c9">s9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><a NAME="E7:23_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</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="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c2">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="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><a NAME="E8:23_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A28</font></span>: 
<font color="Maroon" title="c9">s9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</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="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E6:23_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A27</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="23_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1_2_1_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="E1:23_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> or <font color="Maroon" title="c1">i1</font> <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> <font color="Maroon" title="c5">G1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</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="23_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>,1)</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E4:23_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E8:23_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A28</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:23_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</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="#E2:23_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E1:23_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="ref" href="gobrd11.html#T27" title="GOBRD11:th.27">GOBRD11:27</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="23_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E43">A30</font></span>: 
<font color="Maroon" title="c1">i1</font> <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> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</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>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</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="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E1:23_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="gobrd13.html#T13" target="_self" title="GOBRD13:th.13">Th14</a></span>;<br/><a NAME="E3:23_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">r9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</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="#E5:23_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:23_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,1)</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</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>  &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E4:23_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E8:23_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A28</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:23_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</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="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E1:23_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="ref" href="gobrd11.html#T30" title="GOBRD11:th.30">GOBRD11:30</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><span class="kw">hence </span><a NAME="E10:23_1_1_1_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</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><span class="kw">hence </span><a NAME="E5:23_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</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="23_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:23_1_2"/><span class="lab"><font color="Green" title="E35">A31</font></span>: 
<font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:23_1_2"/><span class="lab"><font color="Green" title="E36">A32</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E3:23_1_2"/><span class="lab"><font color="Green" title="E37">A33</font></span>: 
<font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_2"><span class="lab"><font color="Green" title="E35">A31</font></span></a>, <a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>;<br/><a NAME="E4:23_1_2"/><span class="lab"><font color="Green" title="E38">A34</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_2"><span class="lab"><font color="Green" title="E36">A32</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><a NAME="E5:23_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T51" title="NAT_D:th.51">NAT_D:51</a></span>;<br/><a NAME="E6:23_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A35</font></span>: 
<font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> <font color="Maroon" title="c6">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:23_1_2"/><span class="lab"><font color="Green" title="E40">A36</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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="c4">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_2"><span class="lab"><font color="Green" title="E36">A32</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_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="E1:23_1_2_1_1"/>
( <font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G2</font> or <font color="Maroon" title="c3">i2</font> <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> <font color="Maroon" title="c6">G2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E28">A10</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="23_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_2_1_1_1"/><span class="lab"><font color="Green" title="E41">A37</font></span>: 
<font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,1)</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> <font color="Olive" title="b1">r</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c4">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:23"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E4:23_1_2"><span class="lab"><font color="Green" title="E38">A34</font></span></a>, <a class="txt" href="#E6:23_1_2"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E7:23_1_2"><span class="lab"><font color="Green" title="E40">A36</font></span></a>, <a class="ref" href="gobrd11.html#T29" title="GOBRD11:th.29">GOBRD11:29</a></span>;<br/><a NAME="E3:23_1_2_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r9</font>, <font color="Olive" title="b2">s9</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r9</font>,<font color="Olive" title="b2">s9</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,1)</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> <font color="Olive" title="b1">r9</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Olive" title="b2">s9</font> &amp; <font color="Olive" title="b2">s9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c4">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 ;<br/><a NAME="E4:23_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A38</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,1)</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E14:23"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E1:23_1_2"><span class="lab"><font color="Green" title="E35">A31</font></span></a></span>;<br/><a NAME="E5:23_1_2_1_1_1"/>
<font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:23_1_2_1_1_1"><span class="lab"><font color="Green" title="E41">A37</font></span></a>, <a class="ref" href="gobrd13.html#T10" target="_self" title="GOBRD13:th.10">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E6:23_1_2_1_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</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="#E3:23_1_2"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="txt" href="#E4:23_1_2_1_1_1"><span class="lab"><font color="Green" title="E42">A38</font></span></a>, <a class="ref" href="gobrd11.html#T27" title="GOBRD11:th.27">GOBRD11:27</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="23_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_2_1_1_2"/><span class="lab"><font color="Green" title="E41">A39</font></span>: 
<font color="Maroon" title="c3">i2</font> <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> <font color="Maroon" title="c6">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,1)</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</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>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c4">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:23"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E4:23_1_2"><span class="lab"><font color="Green" title="E38">A34</font></span></a>, <a class="txt" href="#E6:23_1_2"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E7:23_1_2"><span class="lab"><font color="Green" title="E40">A36</font></span></a>, <a class="ref" href="gobrd11.html#T32" title="GOBRD11:th.32">GOBRD11:32</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r9</font>, <font color="Maroon" title="c9">s9</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:23_1_2_1_1_2"/><span class="lab"><font color="Green" title="E42">A40</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c8">r9</font>,<font color="Maroon" title="c9">s9</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E5:23_1_2_1_1_2"/><span class="lab"><font color="Green" title="E43">A41</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,1)</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> <font color="Maroon" title="c8">r9</font>
 <span class="kw">and </span><br/><a NAME="E6:23_1_2_1_1_2"/><span class="lab"><font color="Green" title="E44">A42</font></span>: 
<font color="Maroon" title="c8">r9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</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">and </span><br/><a NAME="E7:23_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Maroon" title="c9">s9</font>
 <span class="kw">and </span><br/><a NAME="E8:23_1_2_1_1_2"/><span class="lab"><font color="Green" title="E45">A43</font></span>: 
<font color="Maroon" title="c9">s9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c4">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><a NAME="E9:23_1_2_1_1_2"/><span class="lab"><font color="Green" title="E46">A44</font></span>: 
( <font color="Maroon" title="c9">s9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,1)</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> <font color="Maroon" title="c8">r9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E14:23"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E1:23_1_2"><span class="lab"><font color="Green" title="E35">A31</font></span></a>, <a class="txt" href="#E5:23_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A41</font></span></a>, <a class="txt" href="#E8:23_1_2_1_1_2"><span class="lab"><font color="Green" title="E45">A43</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</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="23_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_1_2_1_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="E1:23_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> or <font color="Maroon" title="c1">i1</font> <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> <font color="Maroon" title="c5">G1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</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="23_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E47">A45</font></span>: 
<font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>,1)</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">A40</font></span></a>, <a class="txt" href="#E9:23_1_2_1_1_2"><span class="lab"><font color="Green" title="E46">A44</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:23_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</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="#E3:23_1_2"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="txt" href="#E1:23_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E47">A45</font></span></a>, <a class="ref" href="gobrd11.html#T27" title="GOBRD11:th.27">GOBRD11:27</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="23_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E47">A46</font></span>: 
<font color="Maroon" title="c1">i1</font> <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> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_2_1_1_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">i2</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="c6">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_2_1_1_2"><span class="lab"><font color="Green" title="E41">A39</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:23_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A47</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">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="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</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="#E6:23"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E4:23_1_2_1_1_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c1">i1</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="c5">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E47">A46</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:23_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</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="#E13:23"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E6:23_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</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>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</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="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:23_1_2_1_1_2"><span class="lab"><font color="Green" title="E41">A39</font></span></a>, <a class="txt" href="#E1:23_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E47">A46</font></span></a>, <a class="txt" href="#E3:23_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E48">A47</font></span></a>, <a class="ref" href="gobrd13.html#T13" target="_self" title="GOBRD13:th.13">Th14</a></span>;<br/><a NAME="E7:23_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">r9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</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="#E6:23_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:23_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,1)</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</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>  &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">A40</font></span></a>, <a class="txt" href="#E9:23_1_2_1_1_2"><span class="lab"><font color="Green" title="E46">A44</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:23_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</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="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E3:23_1_2"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="txt" href="#E1:23_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E47">A46</font></span></a>, <a class="ref" href="gobrd11.html#T30" title="GOBRD11:th.30">GOBRD11:30</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><span class="kw">hence </span><a NAME="E11:23_1_2_1_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</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><span class="kw">hence </span><a NAME="E9:23_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</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="23_1_3"><span class="kw">suppose </span></a><a NAME="E1:23_1_3"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j1</font> &amp; <font color="Maroon" title="c4">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:23_1_3"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</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="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="gobrd13.html#T11" target="_self" title="GOBRD13:th.11">Th12</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="23_1_4"><span class="kw">suppose </span></a><a NAME="E1:23_1_4"/><span class="lab"><font color="Green" title="E35">A48</font></span>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j1</font> &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A49</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><a NAME="E3:23_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A50</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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="c4">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T43" title="NAT_D:th.43">NAT_D:43</a></span>;<br/><a NAME="E4:23_1_4"/>
<font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_4"><span class="lab"><font color="Green" title="E36">A49</font></span></a>, <a class="ref" href="nat_d.html#T51" title="NAT_D:th.51">NAT_D:51</a></span>;<br/><a NAME="E5:23_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A51</font></span>: 
<font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> <font color="Maroon" title="c6">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:23_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A52</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E2:23_1_4"><span class="lab"><font color="Green" title="E36">A49</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><a NAME="E7:23_1_4"/><span class="lab"><font color="Green" title="E40">A53</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_4"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><a NAME="E8:23_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A54</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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="c2">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T43" title="NAT_D:th.43">NAT_D:43</a></span>;<br/><a NAME="E9:23_1_4"/>
<font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1_4"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="ref" href="nat_d.html#T51" title="NAT_D:th.51">NAT_D:51</a></span>;<br/><a NAME="E10:23_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A55</font></span>: 
<font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:23_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E7:23_1_4"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/><a NAME="E12:23_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A56</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:23_1_4"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="txt" href="#E6:23_1_4"><span class="lab"><font color="Green" title="E39">A52</font></span></a>, <a class="ref" href="gobrd13.html#T16" target="_self" title="GOBRD13:th.16">Th17</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_4_1_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="E1:23_1_4_1_1"/>
( <font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G2</font> or <font color="Maroon" title="c3">i2</font> <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> <font color="Maroon" title="c6">G2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23"><span class="lab"><font color="Green" title="E28">A10</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="23_1_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_4_1_1_1"/><span class="lab"><font color="Green" title="E44">A57</font></span>: 
<font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,1)</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> <font color="Olive" title="b1">r</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c4">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:23"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E2:23_1_4"><span class="lab"><font color="Green" title="E36">A49</font></span></a>, <a class="txt" href="#E5:23_1_4"><span class="lab"><font color="Green" title="E38">A51</font></span></a>, <a class="txt" href="#E3:23_1_4"><span class="lab"><font color="Green" title="E37">A50</font></span></a>, <a class="ref" href="gobrd11.html#T29" title="GOBRD11:th.29">GOBRD11:29</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r9</font>, <font color="Maroon" title="c9">s9</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:23_1_4_1_1_1"/><span class="lab"><font color="Green" title="E45">A58</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c8">r9</font>,<font color="Maroon" title="c9">s9</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E5:23_1_4_1_1_1"/><span class="lab"><font color="Green" title="E46">A59</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,1)</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> <font color="Maroon" title="c8">r9</font>
 <span class="kw">and </span><br/><a NAME="E6:23_1_4_1_1_1"/><span class="lab"><font color="Green" title="E47">A60</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Maroon" title="c9">s9</font> &amp; <font color="Maroon" title="c9">s9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c4">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 ;<br/><a NAME="E7:23_1_4_1_1_1"/><span class="lab"><font color="Green" title="E48">A61</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,1)</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> <font color="Maroon" title="c8">r9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E5:23_1_4_1_1_1"><span class="lab"><font color="Green" title="E46">A59</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E8:23_1_4_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Maroon" title="c9">s9</font> &amp; <font color="Maroon" title="c9">s9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c2">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="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E12:23_1_4"><span class="lab"><font color="Green" title="E43">A56</font></span></a>, <a class="txt" href="#E6:23_1_4_1_1_1"><span class="lab"><font color="Green" title="E47">A60</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:23_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A62</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,1)</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> <font color="Olive" title="b1">r</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c2">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_4_1_1_1"><span class="lab"><font color="Green" title="E45">A58</font></span></a>, <a class="txt" href="#E7:23_1_4_1_1_1"><span class="lab"><font color="Green" title="E48">A61</font></span></a></span>;<br/><a NAME="E10:23_1_4_1_1_1"/>
<font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:23_1_4_1_1_1"><span class="lab"><font color="Green" title="E44">A57</font></span></a>, <a class="ref" href="gobrd13.html#T10" target="_self" title="GOBRD13:th.10">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E11:23_1_4_1_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</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="#E7:23_1_4"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="txt" href="#E10:23_1_4"><span class="lab"><font color="Green" title="E42">A55</font></span></a>, <a class="txt" href="#E8:23_1_4"><span class="lab"><font color="Green" title="E41">A54</font></span></a>, <a class="txt" href="#E9:23_1_4_1_1_1"><span class="lab"><font color="Green" title="E49">A62</font></span></a>, <a class="ref" href="gobrd11.html#T29" title="GOBRD11:th.29">GOBRD11:29</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="23_1_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_4_1_1_2"/><span class="lab"><font color="Green" title="E44">A63</font></span>: 
<font color="Maroon" title="c3">i2</font> <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> <font color="Maroon" title="c6">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,1)</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</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>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c4">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:23"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E2:23_1_4"><span class="lab"><font color="Green" title="E36">A49</font></span></a>, <a class="txt" href="#E5:23_1_4"><span class="lab"><font color="Green" title="E38">A51</font></span></a>, <a class="txt" href="#E3:23_1_4"><span class="lab"><font color="Green" title="E37">A50</font></span></a>, <a class="ref" href="gobrd11.html#T32" title="GOBRD11:th.32">GOBRD11:32</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r9</font>, <font color="Maroon" title="c9">s9</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:23_1_4_1_1_2"/><span class="lab"><font color="Green" title="E45">A64</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c8">r9</font>,<font color="Maroon" title="c9">s9</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E5:23_1_4_1_1_2"/><span class="lab"><font color="Green" title="E46">A65</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i2</font>,1)</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> <font color="Maroon" title="c8">r9</font>
 <span class="kw">and </span><br/><a NAME="E6:23_1_4_1_1_2"/><span class="lab"><font color="Green" title="E47">A66</font></span>: 
<font color="Maroon" title="c8">r9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</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">and </span><br/><a NAME="E7:23_1_4_1_1_2"/><span class="lab"><font color="Green" title="E48">A67</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Maroon" title="c9">s9</font> &amp; <font color="Maroon" title="c9">s9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c4">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 ;<br/><a NAME="E8:23_1_4_1_1_2"/><span class="lab"><font color="Green" title="E49">A68</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,1)</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> <font color="Maroon" title="c8">r9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E11:23"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E5:23_1_4_1_1_2"><span class="lab"><font color="Green" title="E46">A65</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E9:23_1_4_1_1_2"/><span class="lab"><font color="Green" title="E50">A69</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Maroon" title="c9">s9</font> &amp; <font color="Maroon" title="c9">s9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c2">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="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E16:23"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E12:23_1_4"><span class="lab"><font color="Green" title="E43">A56</font></span></a>, <a class="txt" href="#E7:23_1_4_1_1_2"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_4_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_4_1_1_2_1_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="E1:23_1_4_1_1_2_1_1"/>
( <font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> or <font color="Maroon" title="c1">i1</font> <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> <font color="Maroon" title="c5">G1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:23"><span class="lab"><font color="Green" title="E30">A12</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="23_1_4_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_4_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E51">A70</font></span>: 
<font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,1)</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> <font color="Olive" title="b1">r</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c2">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_4_1_1_2"><span class="lab"><font color="Green" title="E45">A64</font></span></a>, <a class="txt" href="#E9:23_1_4_1_1_2"><span class="lab"><font color="Green" title="E50">A69</font></span></a>, <a class="txt" href="#E8:23_1_4_1_1_2"><span class="lab"><font color="Green" title="E49">A68</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:23_1_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</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="#E7:23_1_4"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="txt" href="#E10:23_1_4"><span class="lab"><font color="Green" title="E42">A55</font></span></a>, <a class="txt" href="#E8:23_1_4"><span class="lab"><font color="Green" title="E41">A54</font></span></a>, <a class="txt" href="#E1:23_1_4_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E51">A70</font></span></a>, <a class="ref" href="gobrd11.html#T29" title="GOBRD11:th.29">GOBRD11:29</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="23_1_4_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_4_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E51">A71</font></span>: 
<font color="Maroon" title="c1">i1</font> <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> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span><br/><div class="add"><a NAME="E2:23_1_4_1_1_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">i2</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="c6">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_4_1_1_2"><span class="lab"><font color="Green" title="E44">A63</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:23_1_4_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A72</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">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="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</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="#E6:23"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E4:23_1_4_1_1_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c1">i1</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="c5">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A71</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:23_1_4_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</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="#E13:23"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>;<br/><a NAME="E6:23_1_4_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">i2</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>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</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="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E13:23"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E8:23"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E9:23"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:23_1_4_1_1_2"><span class="lab"><font color="Green" title="E44">A63</font></span></a>, <a class="txt" href="#E1:23_1_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A71</font></span></a>, <a class="txt" href="#E3:23_1_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E52">A72</font></span></a>, <a class="ref" href="gobrd13.html#T13" target="_self" title="GOBRD13:th.13">Th14</a></span>;<br/><a NAME="E7:23_1_4_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">r9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</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="#E6:23_1_4_1_1_2"><span class="lab"><font color="Green" title="E47">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:23_1_4_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c1">i1</font>,1)</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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</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>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</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> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c2">j1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_4_1_1_2"><span class="lab"><font color="Green" title="E45">A64</font></span></a>, <a class="txt" href="#E9:23_1_4_1_1_2"><span class="lab"><font color="Green" title="E50">A69</font></span></a>, <a class="txt" href="#E8:23_1_4_1_1_2"><span class="lab"><font color="Green" title="E49">A68</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:23_1_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</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="#E5:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E7:23_1_4"><span class="lab"><font color="Green" title="E40">A53</font></span></a>, <a class="txt" href="#E10:23_1_4"><span class="lab"><font color="Green" title="E42">A55</font></span></a>, <a class="txt" href="#E8:23_1_4"><span class="lab"><font color="Green" title="E41">A54</font></span></a>, <a class="txt" href="#E1:23_1_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A71</font></span></a>, <a class="ref" href="gobrd11.html#T32" title="GOBRD11:th.32">GOBRD11:32</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><span class="kw">hence </span><a NAME="E11:23_1_4_1_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</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><span class="kw">hence </span><a NAME="E14:23_1_4"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</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>
