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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> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="finseq_2.html#K1">idseq</a> 2 as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2 <b>by </b><i><a class="txt" href="matrix_9.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="setwop_2.html#K2">Seg</a> 2 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="finseq_5.html#K3">Rev</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1">idseq</a> 2</span>)</span> as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2 <b>by </b><i><a class="ref" href="matrix_9.html#T4" target="_self">Th4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="matrix_9.html#K1" target="_self">PPath_product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>
<i><font color="Green">E17</font></i>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i><a class="ref" href="matrix_9.html#T9" target="_self">Th9</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="matrix_9.html#T11" target="_self">Th11</a></i>
;<br/>
<i><font color="Green">E18</font></i>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i><a class="ref" href="matrix_9.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="matrix_9.html#T11" target="_self">Th11</a></i>
;<br/>
<a NAME="E6:23"/>
 <a href="matrix_2.html#K11">Permutations</a> 2 is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T30">MATRIX_2:30</a></i>;<br/>
<a NAME="E7:23"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K11">Permutations</a> 2
 
<b>by </b><i><a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E9:23"/>
1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 2
 
;<br/>
<a NAME="E10:23"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a>, <a class="ref" href="matrix_9.html#T2" target="_self">Th2</a></i>;<br/>
 <a href="matrix_9.html#K2" target="_self">Per</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_9.html#E10:23"><i><font color="Green">E20</font></i></a>, <a class="ref" href="setwop_2.html#T3">SETWOP_2:3</a>, <a class="txt" href="matrix_9.html#E7:23"><i><font color="Green">E19</font></i></a>, <a class="ref" href="matrix_9.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_9.html#E4:23"><i><font color="Green">E17</font></i></a>, <a class="txt" href="matrix_9.html#E5:23"><i><font color="Green">E18</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E12:23"/>
 <a href="matrix_9.html#K2" target="_self">Per</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#K6">][</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>


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