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<b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G1</font> be   <a href="goboard1.html#NM1">Go-board</a>, <font color="Maroon">G2</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E52</font></i>: 
 <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">G1</font> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">G2</font>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G1</font>
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G2</font>
 <b>and </b><br/><a NAME="E4:24"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:24"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G2</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/>

<a NAME="E6:24"/><i><font color="Green">E62</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E7:24"/><i><font color="Green">E63</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E8:24"/><b>then </b><i><font color="Green">E64</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="ref" href="goboard5.html#T2">GOBOARD5:2</a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> ) or ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1 ) or ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:24_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
( <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_1_1"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font> or <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="gobrd11.html#T25">GOBRD11:25</a></i>;<br/><b>then consider </b><font color="Maroon">r'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:24_1_1_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r'</font>,<font color="Maroon">s'</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:24_1_1_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>and </b><br/><a NAME="E6:24_1_1_1_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s'</font>
 ;<br/><a NAME="E7:24_1_1_1_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i>, , , , , </i>;<br/><a NAME="E8:24_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E9:24_1_1_1_1_1"/><b>then </b>
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E10:24_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) </span>}</span> 
 <b>by </b><i>, , , , , </i>;<br/><b>hence </b><a NAME="E11:24_1_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 <b>by </b><i>, , <a class="ref" href="gobrd11.html#T25">GOBRD11:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i>, , , <a class="ref" href="gobrd11.html#T26">GOBRD11:26</a></i>;<br/><b>then consider </b><font color="Maroon">r'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:24_1_1_1_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r'</font>,<font color="Maroon">s'</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:24_1_1_1_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>and </b><br/><a NAME="E6:24_1_1_1_1_2"/><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s'</font>
 <b>and </b><br/><a NAME="E7:24_1_1_1_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">s'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E8:24_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E9:24_1_1_1_1_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_1_1_2_1_1"/>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> or <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_2_1_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) </span>}</span> 
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:24_1_1_1_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 <b>by </b><i>, , <a class="ref" href="gobrd11.html#T25">GOBRD11:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_2_1_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_1_2_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:24_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">G1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:24_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , , , , , </i>;<br/><a NAME="E5:24_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">s'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:24_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E7:24_1_1_1_1_2_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 <b>by </b><i>, , , <a class="ref" href="gobrd11.html#T26">GOBRD11:26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E11:24_1_1_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:24_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E2:24_1_2"/><i><font color="Green">E102</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><a NAME="E3:24_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><a NAME="E4:24_1_2"/><i><font color="Green">E104</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E5:24_1_2"/><b>then </b>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T14">JORDAN3:14</a></i>;<br/><a NAME="E6:24_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:24_1_2"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_2_1_1"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font> or <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font>
 ;<br/><div class="add"><a NAME="E2:24_1_2_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) </span>}</span> 
 <b>by </b><i>, , , , <a class="ref" href="gobrd11.html#T31">GOBRD11:31</a></i>;<br/><b>then consider </b><font color="Maroon">r'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:24_1_2_1_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r'</font>,<font color="Maroon">s'</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:24_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font>
 <b>and </b><br/><a NAME="E6:24_1_2_1_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>and </b><br/><a NAME="E7:24_1_2_1_1_1"/><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s'</font>
 ;<br/><a NAME="E8:24_1_2_1_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i>, , , , , </i>;<br/><a NAME="E9:24_1_2_1_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E10:24_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s'</font>
 <b>by </b><i>, , , , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E11:24_1_2_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E12:24_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 <b>by </b><i>, , <a class="ref" href="gobrd11.html#T25">GOBRD11:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font>
 ;<br/><div class="add"><a NAME="E2:24_1_2_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i>, , , , , <a class="ref" href="gobrd11.html#T32">GOBRD11:32</a></i>;<br/><b>then consider </b><font color="Maroon">r'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:24_1_2_1_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r'</font>,<font color="Maroon">s'</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:24_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font>
 <b>and </b><br/><a NAME="E6:24_1_2_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>and </b><br/><a NAME="E7:24_1_2_1_1_2"/><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s'</font>
 <b>and </b><br/><a NAME="E8:24_1_2_1_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">s'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E9:24_1_2_1_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E10:24_1_2_1_1_2"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s'</font>
 <b>by </b><i>, , , , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_2_1_1_2_1_1"/>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> or <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_2_1_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:24_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:24_1_2_1_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 <b>by </b><i>, , <a class="ref" href="gobrd11.html#T25">GOBRD11:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_2_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:24_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:24_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">G1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:24_1_2_1_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E5:24_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , , , , , </i>;<br/><a NAME="E6:24_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">s'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:24_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E8:24_1_2_1_1_2_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 <b>by </b><i>, , , <a class="ref" href="gobrd11.html#T26">GOBRD11:26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:24_1_2_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:24_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_3"><b>suppose </b></a><a NAME="E1:24_1_3"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_4"><b>suppose </b></a><a NAME="E1:24_1_4"/><i><font color="Green">E125</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> )
 ;<br/><div class="add"><a NAME="E2:24_1_4"/><b>then </b><i><font color="Green">E126</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 )
 <b>by </b><i><a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E3:24_1_4"/><b>then </b>
( <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> )
 <b>by </b><i><a class="ref" href="jordan3.html#T14">JORDAN3:14</a></i>;<br/><a NAME="E4:24_1_4"/><b>then </b><i><font color="Green">E127</font></i>: 
( <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G2</font> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:24_1_4"/><i><font color="Green">E128</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> )
 <b>by </b><i>, <a class="ref" href="jordan3.html#T6">JORDAN3:6</a></i>;<br/><a NAME="E6:24_1_4"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">G1</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E7:24_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , , , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E8:24_1_4"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , , , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_4_1_1"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font> or <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_4_1_1_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font>
 ;<br/><div class="add"><a NAME="E2:24_1_4_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) </span>}</span> 
 <b>by </b><i>, , , , <a class="ref" href="gobrd11.html#T31">GOBRD11:31</a></i>;<br/><b>then consider </b><font color="Maroon">r'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:24_1_4_1_1_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r'</font>,<font color="Maroon">s'</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:24_1_4_1_1_1"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font>
 <b>and </b><br/><a NAME="E6:24_1_4_1_1_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>and </b><br/><a NAME="E7:24_1_4_1_1_1"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s'</font>
 ;<br/><a NAME="E8:24_1_4_1_1_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i>, , , , , </i>;<br/><a NAME="E9:24_1_4_1_1_1"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:24_1_4_1_1_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E11:24_1_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s'</font>
 <b>by </b><i>, , , , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E12:24_1_4_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E13:24_1_4_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 <b>by </b><i>, , , , <a class="ref" href="gobrd11.html#T31">GOBRD11:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_4_1_1_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font>
 ;<br/><div class="add"><a NAME="E2:24_1_4_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i>, , , , , <a class="ref" href="gobrd11.html#T32">GOBRD11:32</a></i>;<br/><b>then consider </b><font color="Maroon">r'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:24_1_4_1_1_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r'</font>,<font color="Maroon">s'</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:24_1_4_1_1_2"/><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font>
 <b>and </b><br/><a NAME="E6:24_1_4_1_1_2"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>and </b><br/><a NAME="E7:24_1_4_1_1_2"/><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s'</font>
 <b>and </b><br/><a NAME="E8:24_1_4_1_1_2"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">s'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E9:24_1_4_1_1_2"/><i><font color="Green">E145</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:24_1_4_1_1_2"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">r'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E11:24_1_4_1_1_2"/><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s'</font>
 <b>by </b><i>, , , , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_4_1_1_2_1_1"/>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> or <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_4_1_1_2_1_1_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:24_1_4_1_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) </span>}</span> 
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E3:24_1_4_1_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 <b>by </b><i>, , , , <a class="ref" href="gobrd11.html#T31">GOBRD11:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_4_1_1_2_1_1_2"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:24_1_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E150</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G2</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:24_1_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">G1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:24_1_4_1_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E5:24_1_4_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , , , , , </i>;<br/><a NAME="E6:24_1_4_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">s'</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:24_1_4_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E8:24_1_4_1_1_2_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd11.html#T32">GOBRD11:32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:24_1_4_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:24_1_4"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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