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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <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>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="matrix_1.html#NM5">Matrix</a> of 3,<font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 3;<br/>

<b>assume </b><a NAME="E2:31"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3">*&gt;</a></span>
 ;<br/>

<a NAME="E3:31"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i><a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E6:31"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:31"/><i><font color="Green">E30</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E6:31"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E8:31"/><i><font color="Green">E31</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E2:31"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E9:31"/><i><font color="Green">E32</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E6:31"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E10:31"/><i><font color="Green">E33</font></i>: 
3 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E2:31"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E11:31"/><i><font color="Green">E34</font></i>: 
3 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E6:31"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E12:31"/><i><font color="Green">E35</font></i>: 
2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E2:31"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E13:31"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1:31"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E14:31"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</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>11</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E3:31"><i><font color="Green">E26</font></i></a>, <a class="txt" href="matrix_9.html#E4:31"><i><font color="Green">E27</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E16:31"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="matrix_1.html#K3">*</a> 1,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E17:31"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E13:31"><i><font color="Green">E36</font></i></a>, <a class="txt" href="matrix_9.html#E16:31"><i><font color="Green">E37</font></i></a></i>;<br/>
<i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">c<sub>11</sub></font> <a href="matrix_1.html#K3">*</a> 1,1
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a>, <a class="txt" href="matrix_9.html#E7:31"><i><font color="Green">E30</font></i></a>, <a class="txt" href="matrix_9.html#E8:31"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="matrix_9.html#E17:31"><i><font color="Green">E38</font></i></a>, <a class="txt" href="matrix_9.html#E16:31"><i><font color="Green">E37</font></i></a></i>
;<br/>
<a NAME="E19:31"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1:31"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E20:31"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,3</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>11</sub></font>
 
<b>by </b><i><a class="ref" href="goboard7.html#T10">GOBOARD7:10</a>, <a class="txt" href="matrix_9.html#E3:31"><i><font color="Green">E26</font></i></a>, <a class="txt" href="matrix_9.html#E4:31"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E22:31"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 2 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="matrix_1.html#K3">*</a> 2,3 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 3 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E23:31"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E19:31"><i><font color="Green">E40</font></i></a>, <a class="txt" href="matrix_9.html#E22:31"><i><font color="Green">E41</font></i></a></i>;<br/>
<i><font color="Green">E43</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 2 = 
<font color="Maroon">c<sub>11</sub></font> <a href="matrix_1.html#K3">*</a> 2,3
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a>, <a class="txt" href="matrix_9.html#E9:31"><i><font color="Green">E32</font></i></a>, <a class="txt" href="matrix_9.html#E10:31"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="matrix_9.html#E23:31"><i><font color="Green">E42</font></i></a>, <a class="txt" href="matrix_9.html#E22:31"><i><font color="Green">E41</font></i></a></i>
;<br/>
<a NAME="E25:31"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1:31"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E26:31"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,2</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>11</sub></font>
 
<b>by </b><i><a class="ref" href="goboard7.html#T10">GOBOARD7:10</a>, <a class="txt" href="matrix_9.html#E3:31"><i><font color="Green">E26</font></i></a>, <a class="txt" href="matrix_9.html#E4:31"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E28:31"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 3 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="matrix_1.html#K3">*</a> 3,2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> 2 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E29:31"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E25:31"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrix_9.html#E28:31"><i><font color="Green">E45</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 3 = 
<font color="Maroon">c<sub>11</sub></font> <a href="matrix_1.html#K3">*</a> 3,2
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a>, <a class="txt" href="matrix_9.html#E11:31"><i><font color="Green">E34</font></i></a>, <a class="txt" href="matrix_9.html#E12:31"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="matrix_9.html#E29:31"><i><font color="Green">E46</font></i></a>, <a class="txt" href="matrix_9.html#E28:31"><i><font color="Green">E45</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E31:31"/>
 <a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="matrix_9.html#E18:31"><i><font color="Green">E39</font></i></a>, <a class="txt" href="matrix_9.html#E24:31"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E5:31"><i><font color="Green">E28</font></i></a></i>;<br/>


</div>
