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<div class="add">

<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="E1:61"/><i><font color="Green">E49</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>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="finseq_2.html#K1">idseq</a> 3 as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="setwop_2.html#K2">Seg</a> 3 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="finseq_2.html#K1">idseq</a> 3 as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 3 <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>14</sub></font> =  <a href="finseq_5.html#K3">Rev</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1">idseq</a> 3</span>)</span> as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 3 <b>by </b><i><a class="ref" href="matrix_9.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:61"/><i><font color="Green">E50</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">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,2,1</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T62">FINSEQ_2:62</a>, <a class="ref" href="matrix_9.html#T15" target="_self">Th15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <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>, <font color="Maroon">c<sub>16</sub></font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3">*&gt;</a></span>, <font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">2,1,3</span><a href="finseq_4.html#K3">*&gt;</a></span>, <font color="Maroon">c<sub>18</sub></font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 3 <b>by </b><i><a class="ref" href="matrix_9.html#T27" target="_self">Th27</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>19</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>20</sub></font> =  <a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font>;<br/>
<i><font color="Green">E51</font></i>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</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">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</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="ref" href="matrix_9.html#T20" target="_self">Th20</a>, <a class="txt" href="matrix_9.html#E1:61"><i><font color="Green">E49</font></i></a>, <a class="txt" href="matrix_9.html#E5:61"><i><font color="Green">E50</font></i></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>6</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="matrix_9.html#T26" target="_self">Th26</a></i>
;<br/>
<i><font color="Green">E52</font></i>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</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">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="matrix_9.html#T21" target="_self">Th21</a>, <a class="txt" href="matrix_9.html#E1:61"><i><font color="Green">E49</font></i></a>, <a class="txt" href="matrix_9.html#E5:61"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="matrix_9.html#T26" target="_self">Th26</a></i>
;<br/>
<i><font color="Green">E53</font></i>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</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">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1"><a href="finseq_4.html#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="ref" href="matrix_9.html#T22" target="_self">Th22</a>, <a class="txt" href="matrix_9.html#E1:61"><i><font color="Green">E49</font></i></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>7</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="matrix_9.html#T26" target="_self">Th26</a></i>
;<br/>
<i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</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">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="matrix_9.html#T23" target="_self">Th23</a>, <a class="txt" href="matrix_9.html#E1:61"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<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>7</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="matrix_9.html#T26" target="_self">Th26</a></i>
;<br/>
<i><font color="Green">E55</font></i>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</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">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</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="ref" href="matrix_9.html#T24" target="_self">Th24</a>, <a class="txt" href="matrix_9.html#E1:61"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<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>5</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="matrix_9.html#T26" target="_self">Th26</a></i>
;<br/>
<i><font color="Green">E56</font></i>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</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">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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="ref" href="matrix_9.html#T25" target="_self">Th25</a>, <a class="txt" href="matrix_9.html#E1:61"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="matrix_9.html#T26" target="_self">Th26</a></i>
;<br/>
<a NAME="E13:61"/>
 <a href="matrix_2.html#K11">Permutations</a> 3 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="E14:61"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K11">Permutations</a> 3
 
<b>by </b><i><a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<a NAME="E15:61"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K11">Permutations</a> 3
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>24</sub></font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>25</sub></font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>26</sub></font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="enumset1.html#K4">}</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 3</span>)</span> <b>by </b><i><a class="txt" href="matrix_9.html#E14:61"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrix_9.html#E5:61"><i><font color="Green">E50</font></i></a>, <a class="ref" href="matrix_9.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E18:61"/><i><font color="Green">E58</font></i>: 
 <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E5:61"><i><font color="Green">E50</font></i></a>, <a class="ref" href="matrix_9.html#T19" target="_self">Th19</a>, <a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<a NAME="E19:61"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#T53">ENUMSET1:53</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="setwiseo.html#K4">.}</a></span>, <font color="Maroon">c<sub>29</sub></font> = <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="setwiseo.html#K4">.}</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 3</span>)</span> ;<br/>
<a NAME="E21:61"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>28</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>29</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_7"><b>proof </b></a><div class="add">

<a NAME="E1:61_7"/>
<font color="Maroon">c<sub>28</sub></font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_7_1"><b>proof </b></a><div class="add">

<a NAME="E1:61_7_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">c<sub>29</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E2:61_7_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">c<sub>29</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>28</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_7_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_7_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font>
 ;<br/><a NAME="E2:61_7_1_1"/>
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> or <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="txt" href="matrix_9.html#E1:61_7_1_1"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E3:61_7_1_1"/><b>then </b>
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font> &amp; not <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="txt" href="matrix_9.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="txt" href="matrix_9.html#E5:61"><i><font color="Green">E50</font></i></a></i>;<br/><b>hence </b><a NAME="E4:61_7_1_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:61_7_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>28</sub></font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:61_7_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="matrix_9.html#E2:61_7_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:61_7"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>28</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E61</font></i>: the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>28</sub></font>,<span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</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#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></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> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="setwop_2.html#T5">SETWOP_2:5</a>, <a class="txt" href="matrix_9.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="txt" href="matrix_9.html#E5:61"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>23</sub></font>

;<br/>
<i><font color="Green">E62</font></i>: the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>29</sub></font>,<span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</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#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></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> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="setwop_2.html#T5">SETWOP_2:5</a>, <a class="txt" href="matrix_9.html#E35"><i><font color="Green">Lemma29</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>26</sub></font>

;<br/>
 <a href="matrix_9.html#K2" target="_self">Per</a> <font color="Maroon">c<sub>11</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>28</sub></font>,<span class="p2">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="setwop_2.html#T6">SETWOP_2:6</a>, <a class="txt" href="matrix_9.html#E19:61"><i><font color="Green">E59</font></i></a>, <a class="txt" href="matrix_9.html#E21:61"><i><font color="Green">E60</font></i></a>, <a class="txt" href="matrix_9.html#E18:61"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_9.html#E22:61"><i><font color="Green">E61</font></i></a>, <a class="txt" href="matrix_9.html#E23:61"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_9.html#E7:61"><i><font color="Green">E51</font></i></a>, <a class="txt" href="matrix_9.html#E8:61"><i><font color="Green">E52</font></i></a>, <a class="txt" href="matrix_9.html#E9:61"><i><font color="Green">E53</font></i></a>, <a class="txt" href="matrix_9.html#E10:61"><i><font color="Green">E54</font></i></a>, <a class="txt" href="matrix_9.html#E11:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrix_9.html#E12:61"><i><font color="Green">E56</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E25:61"/>
 <a href="matrix_9.html#K2" target="_self">Per</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/>


</div>
