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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green">E32</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="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E3:42"/><i><font color="Green">E34</font></i>: 
 <a href="gobrd13.html#K2" target="_self">Values</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="ref" href="gobrd13.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E4:42"/>
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="goboard5.html#D5">GOBOARD5:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <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="E6:42"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:42"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E8:42"/><i><font color="Green">E37</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E9:42"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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="E10:42"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E11:42"/><i><font color="Green">E39</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E35</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E12:42"/><i><font color="Green">E40</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &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> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E37</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E13:42"/><b>then </b><i><font color="Green">E41</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/>
<a NAME="E14:42"/><i><font color="Green">E42</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:42"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</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>2</sub></font>
 <b>and </b><br/><a NAME="E17:42"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E18:42"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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>2</sub></font>
 <b>and </b><br/><a NAME="E19:42"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E20:42"/><i><font color="Green">E47</font></i>: 
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E32</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E21:42"/><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E22:42"/><i><font color="Green">E49</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E18:42"><i><font color="Green">E45</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E23:42"/><b>then </b><i><font color="Green">E50</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/>
<a NAME="E24:42"/><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/>
<a NAME="E25:42"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E26:42"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E27:42"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="gobrd13.html#E20:42"><i><font color="Green">E47</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>suppose </b></a><a NAME="E1:42_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:42_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E42</font></i></a>, <a class="ref" href="sprect_3.html#T24">SPRECT_3:24</a></i>;<br/><a NAME="E3:42_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:42_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E24:42"><i><font color="Green">E51</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div><a NAME="E3:42_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:42_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:42_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:42_1_1_2"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan5d.html#T10">JORDAN5D:10</a></i>;<br/><a NAME="E7:42_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E5:42_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E8:42_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:42_1_1_2"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</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>2</sub></font>
 <b>and </b><br/><a NAME="E11:42_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E4:42_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E12:42_1_1_2"/><i><font color="Green">E65</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E10:42_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E13:42_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E14:42_1_1_2"/><i><font color="Green">E67</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E12:42_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E2:42_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="txt" href="gobrd13.html#E24:42"><i><font color="Green">E51</font></i></a>, <a class="txt" href="gobrd13.html#E12:42_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="sprect_3.html#T24">SPRECT_3:24</a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd13.html#E8:42_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E11:42_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E2:42_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E12:42_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="sprect_3.html#T24">SPRECT_3:24</a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E42</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd13.html#E8:42_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E11:42_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:42_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd13.html#E15:42_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><a NAME="E5:42_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:42_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp;  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E32</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#D2" target="_self">Def2</a>, <a class="ref" href="goboard5.html#D6">GOBOARD5:def 6</a></i>;<br/><a NAME="E7:42_1_1"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E32</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd13.html#E5:42_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="gobrd13.html#D3" target="_self">Def3</a>, <a class="ref" href="goboard5.html#D7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E8:42_1_1"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42"><i><font color="Green">E34</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="gobrd13.html#T18" target="_self">Th18</a>, <a class="ref" href="gobrd13.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>suppose </b></a><a NAME="E1:42_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:42_1_2_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green">E41</font></i></a>, <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/><a NAME="E3:42_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:42_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><a NAME="E3:42_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_2_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:42_1_2_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:42_1_2_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:42_1_2_2"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>14</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_2_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan5d.html#T9">JORDAN5D:9</a></i>;<br/><a NAME="E7:42_1_2_2"/><i><font color="Green">E62</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:42_1_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E5:42_1_2_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E9:42_1_2_2"/><b>then </b><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E10:42_1_2_2"/><i><font color="Green">E64</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:42_1_2_2"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</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>2</sub></font>
 <b>and </b><br/><a NAME="E13:42_1_2_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E4:42_1_2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E14:42_1_2_2"/><i><font color="Green">E67</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42_1_2_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E15:42_1_2_2"/><b>then </b><i><font color="Green">E68</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E16:42_1_2_2"/><i><font color="Green">E69</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_2_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_2_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:42_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_2_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E15:42_1_2_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/><b>hence </b><a NAME="E3:42_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_2_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_2_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd13.html#E9:42_1_2_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_2_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_2_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:42_1_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_2_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E16:42_1_2_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/><b>hence </b><a NAME="E3:42_1_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_2_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_2_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd13.html#E7:42_1_2_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E9:42_1_2_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E10:42_1_2_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_2_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:42_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd13.html#E17:42_1_2_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><a NAME="E5:42_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:42_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> &amp;  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E32</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#D2" target="_self">Def2</a>, <a class="ref" href="goboard5.html#D6">GOBOARD5:def 6</a></i>;<br/><a NAME="E7:42_1_2"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E32</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd13.html#E5:42_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="gobrd13.html#D3" target="_self">Def3</a>, <a class="ref" href="goboard5.html#D7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E8:42_1_2"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42"><i><font color="Green">E34</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="gobrd13.html#T18" target="_self">Th18</a>, <a class="ref" href="gobrd13.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_3"><b>suppose </b></a><a NAME="E1:42_1_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:42_1_3_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green">E41</font></i></a>, <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/><a NAME="E3:42_1_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:42_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_3_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><a NAME="E3:42_1_3"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:42_1_3_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:42_1_3_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:42_1_3_2"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>14</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_3_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan5d.html#T9">JORDAN5D:9</a></i>;<br/><a NAME="E7:42_1_3_2"/><i><font color="Green">E62</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:42_1_3_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E5:42_1_3_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E9:42_1_3_2"/><b>then </b><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_3_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E10:42_1_3_2"/><i><font color="Green">E64</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_3_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:42_1_3_2"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</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>2</sub></font>
 <b>and </b><br/><a NAME="E13:42_1_3_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E4:42_1_3_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E14:42_1_3_2"/><i><font color="Green">E67</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42_1_3_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E15:42_1_3_2"/><b>then </b><i><font color="Green">E68</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E16:42_1_3_2"/><i><font color="Green">E69</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_3_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:42_1_3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_3_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E16:42_1_3_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/><b>hence </b><a NAME="E3:42_1_3_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_3_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_3_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd13.html#E9:42_1_3_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_3_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:42_1_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_3_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E15:42_1_3_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/><b>hence </b><a NAME="E3:42_1_3_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_3_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_3_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd13.html#E7:42_1_3_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E9:42_1_3_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E10:42_1_3_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_3_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:42_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd13.html#E17:42_1_3_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><a NAME="E5:42_1_3"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42_1_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:42_1_3"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp;  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E32</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#D2" target="_self">Def2</a>, <a class="ref" href="goboard5.html#D6">GOBOARD5:def 6</a></i>;<br/><a NAME="E7:42_1_3"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E32</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd13.html#E5:42_1_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="gobrd13.html#D3" target="_self">Def3</a>, <a class="ref" href="goboard5.html#D7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E8:42_1_3"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42"><i><font color="Green">E34</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="gobrd13.html#T18" target="_self">Th18</a>, <a class="ref" href="gobrd13.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4"><b>suppose </b></a><a NAME="E1:42_1_4"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_4_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_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="E1:42_1_4_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 </span><b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_4_1"><i><font color="Green">E57</font></i></a>, <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="42_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_4_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green">E41</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E23:42"><i><font color="Green">E50</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_4_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green">E41</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E23:42"><i><font color="Green">E50</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_4_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:42_1_4_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E42</font></i></a>, <a class="ref" href="sprect_3.html#T24">SPRECT_3:24</a></i>;<br/><a NAME="E3:42_1_4_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:42_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E24:42"><i><font color="Green">E51</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_4_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div><a NAME="E4:42_1_4"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_4_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:42_1_4_3"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:42_1_4_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:42_1_4_3"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_4_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan5d.html#T10">JORDAN5D:10</a></i>;<br/><a NAME="E7:42_1_4_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E5:42_1_4_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E8:42_1_4_3"/><b>then </b><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_4_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E9:42_1_4_3"/><i><font color="Green">E64</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_4_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:42_1_4_3"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</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>2</sub></font>
 <b>and </b><br/><a NAME="E12:42_1_4_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E4:42_1_4_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E13:42_1_4_3"/><i><font color="Green">E67</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42_1_4_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E14:42_1_4_3"/><b>then </b><i><font color="Green">E68</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E15:42_1_4_3"/><i><font color="Green">E69</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_4_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_4_3_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E2:42_1_4_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green">E49</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_4_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E15:42_1_4_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="sprect_3.html#T24">SPRECT_3:24</a></i>;<br/><b>hence </b><a NAME="E3:42_1_4_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_4_3"><i><font color="Green">E59</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_4_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E8:42_1_4_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E12:42_1_4_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_4_3_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E2:42_1_4_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green">E48</font></i></a>, <a class="txt" href="gobrd13.html#E24:42"><i><font color="Green">E51</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_4_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_4_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sprect_3.html#T24">SPRECT_3:24</a></i>;<br/><b>hence </b><a NAME="E3:42_1_4_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4_3"><i><font color="Green">E58</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_4_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E8:42_1_4_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E9:42_1_4_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="gobrd13.html#E12:42_1_4_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:42_1_4_3"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd13.html#E16:42_1_4_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><a NAME="E6:42_1_4"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E4:42_1_4"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E7:42_1_4"/><b>then </b><i><font color="Green">E59</font></i>: 
(  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> &amp;  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E32</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#D2" target="_self">Def2</a>, <a class="ref" href="goboard5.html#D6">GOBOARD5:def 6</a></i>;<br/><a NAME="E8:42_1_4"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E32</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green">E53</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_4"><i><font color="Green">E58</font></i></a>, <a class="ref" href="gobrd13.html#D3" target="_self">Def3</a>, <a class="ref" href="goboard5.html#D7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E9:42_1_4"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42"><i><font color="Green">E34</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E38</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green">E46</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_4"><i><font color="Green">E56</font></i></a>, <a class="txt" href="gobrd13.html#E7:42_1_4"><i><font color="Green">E59</font></i></a>, <a class="ref" href="gobrd13.html#T18" target="_self">Th18</a>, <a class="ref" href="gobrd13.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
