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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green">E8</font></i>: 
 <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E9</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:14"/><i><font color="Green">E10</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E4:14"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E5:14"/><i><font color="Green">E12</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E2:14"><i><font color="Green">E9</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E6:14"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E7:14"/><b>then </b><i><font color="Green">E13</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E8:14"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E9:14"/><b>then </b><i><font color="Green">E14</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E10:14"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E11:14"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gobrd13.html#E5:14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E12:14"/><b>then </b><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="gobrd13.html#E5:14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/>
<a NAME="E13:14"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="gobrd13.html#E5:14"><i><font color="Green">E12</font></i></a>, <a class="txt" href="gobrd13.html#E9:14"><i><font color="Green">E14</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E14:14"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>4</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E15:14"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="gobrd13.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E16:14"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="gobrd13.html#E1:14"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E17:14"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>5</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="gobrd13.html#T7" target="_self">Th7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:14"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E20:14"/><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font>
 ;<br/>
<a NAME="E21:14"/><i><font color="Green">E18</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E20:14"><i><font color="Green">E17</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<b>then </b><i><font color="Green">E19</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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="txt" href="gobrd13.html#E7:14"><i><font color="Green">E13</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="gobrd13.html#E3:14"><i><font color="Green">E10</font></i></a>, <a class="txt" href="gobrd13.html#E7:14"><i><font color="Green">E13</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/>
<a NAME="E23:14"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="gobrd13.html#E4:14"><i><font color="Green">E11</font></i></a>, <a class="txt" href="gobrd13.html#E12:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="gobrd13.html#E19:14"><i><font color="Green">E16</font></i></a>, <a class="txt" href="gobrd13.html#E21:14"><i><font color="Green">E18</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E24:14"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E4:14"><i><font color="Green">E11</font></i></a>, <a class="txt" href="gobrd13.html#E12:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="gobrd13.html#E19:14"><i><font color="Green">E16</font></i></a>, <a class="txt" href="gobrd13.html#E21:14"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gobrd13.html#E22:14"><i><font color="Green">E19</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/>


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