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<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">e</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM5">Matrix</a> of 3,<font color="Maroon">K</font>;<br/>





<b>assume </b><a NAME="E1:61"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">M</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">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</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">d</font>,<font color="Maroon">e</font>,<font color="Maroon">f</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">g</font>,<font color="Maroon">h</font>,<font color="Maroon">i</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">id3</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">Id3</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#K12">Permutations</a> 3 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">rid3</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#K12">Permutations</a> 3 <b>by </b><i/>;<br/>
<a NAME="E5:61"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">id3</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">rid3</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>, </i>;<br/>
<b>reconsider </b><font color="Maroon">a3</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">a4</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">a5</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">a6</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#K12">Permutations</a> 3 <b>by </b><i><a class="ref" href="matrix_9.html#T1" target="_self">Th1</a></i>;<br/>
<b>set </b><font color="Maroon">F</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font>;<br/>
<b>set </b><font color="Maroon">r</font> =  <a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font>;<br/>
<a NAME="E7:61"/><i><font color="Green">E37</font></i>: 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 3</span>)</span> is <a href="matrix_2.html#V3">even</a>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T29">MATRIX_2:29</a></i>;<br/>
<a NAME="E8:61"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">id3</font> is <a href="matrix_2.html#V3">even</a>
 
<b>by </b><i/>;<br/>
<a NAME="E9:61"/><i><font color="Green">E39</font></i>: 
3 <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K11">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> 3</span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T20">MATRIX_2:20</a></i>;<br/>
<a NAME="E10:61"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">Id3</font> is <a href="matrix_2.html#V3">even</a>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E6"><i><font color="Green">Lemma53</font></i></a></i>;<br/>
<i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">id3</font> = 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">Id3</font>,<font color="Maroon">M</font></span>)</span></span>)</span>,<font color="Maroon">Id3</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">Id3</font>,<font color="Maroon">M</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a>, <a class="ref" href="matrix_9.html#T2" target="_self">Th2</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</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">a</font>,<font color="Maroon">e</font>,<font color="Maroon">i</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">e</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="matrix_9.html#E1"><i><font color="Green">Lemma27</font></i></a></i>
;<br/>
<a NAME="E12:61"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">rid3</font> is <a href="matrix_2.html#V3">even</a>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E4"><i><font color="Green">Lemma47</font></i></a>, , </i>;<br/>
<i><font color="Green">E43</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">rid3</font> = 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">rid3</font>,<font color="Maroon">M</font></span>)</span></span>)</span>,<font color="Maroon">rid3</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">rid3</font>,<font color="Maroon">M</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#T4" target="_self">Th4</a>, <a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">e</font>,<font color="Maroon">g</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K10">*</a> <font color="Maroon">e</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_9.html#E1"><i><font color="Green">Lemma27</font></i></a></i>
;<br/>
<a NAME="E14:61"/><i><font color="Green">E44</font></i>: 
not <font color="Maroon">a3</font> is <a href="matrix_2.html#V3">even</a>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E45</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a3</font> = 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">a3</font>,<font color="Maroon">M</font></span>)</span></span>)</span>,<font color="Maroon">a3</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">a3</font>,<font color="Maroon">M</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#T6" target="_self">Th6</a>, <a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">f</font>,<font color="Maroon">h</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">f</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_9.html#E1"><i><font color="Green">Lemma27</font></i></a></i>
;<br/>
<a NAME="E16:61"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">a4</font> is <a href="matrix_2.html#V3">even</a>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E11:61"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<i><font color="Green">E73</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a4</font> = 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">a4</font>,<font color="Maroon">M</font></span>)</span></span>)</span>,<font color="Maroon">a4</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">a4</font>,<font color="Maroon">M</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</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">b</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">f</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font>
<b>by </b><i><a class="txt" href="matrix_9.html#E1"><i><font color="Green">Lemma27</font></i></a></i>
;<br/>
<a NAME="E18:61"/><i><font color="Green">E74</font></i>: 
not <font color="Maroon">a5</font> is <a href="matrix_2.html#V3">even</a>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E5"><i><font color="Green">Lemma48</font></i></a>, </i>;<br/>
<i><font color="Green">E75</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a5</font> = 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">a5</font>,<font color="Maroon">M</font></span>)</span></span>)</span>,<font color="Maroon">a5</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">a5</font>,<font color="Maroon">M</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#D1" target="_self">Def1</a>, <a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font>,<font color="Maroon">i</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_9.html#E1"><i><font color="Green">Lemma27</font></i></a></i>
;<br/>
<a NAME="E20:61"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">a6</font> is <a href="matrix_2.html#V3">even</a>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E12:61"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<i><font color="Green">E78</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a6</font> = 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">a6</font>,<font color="Maroon">M</font></span>)</span></span>)</span>,<font color="Maroon">a6</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">a6</font>,<font color="Maroon">M</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</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</font>,<font color="Maroon">d</font>,<font color="Maroon">h</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">h</font>
<b>by </b><i><a class="txt" href="matrix_9.html#E1"><i><font color="Green">Lemma27</font></i></a></i>
;<br/>
<a NAME="E22:61"/>
 <a href="matrix_2.html#K12">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="E23:61"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> 3</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K12">Permutations</a> 3
 
<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">r1</font> = <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">id3</font>, <font color="Maroon">r2</font> = <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">rid3</font>, <font color="Maroon">r3</font> = <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a3</font>, <font color="Maroon">r4</font> = <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a4</font>, <font color="Maroon">r5</font> = <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a5</font>, <font color="Maroon">r6</font> = <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a6</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i><a class="ref" href="matrix_9.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E25:61"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><font color="Maroon">Id3</font>,<font color="Maroon">rid3</font>,<font color="Maroon">a3</font>,<font color="Maroon">a4</font>,<font color="Maroon">a5</font>,<font color="Maroon">a6</font></span><a href="enumset1.html#K4">}</a></span> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K12">Permutations</a> 3
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><font color="Maroon">Id3</font>,<font color="Maroon">rid3</font>,<font color="Maroon">a3</font>,<font color="Maroon">a4</font>,<font color="Maroon">a5</font>,<font color="Maroon">a6</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#K12">Permutations</a> 3</span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E27:61"/><i><font color="Green">E82</font></i>: 
 <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<a NAME="E28:61"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">Id3</font>,<font color="Maroon">rid3</font>,<font color="Maroon">a3</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">a4</font>,<font color="Maroon">a5</font>,<font color="Maroon">a6</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">B1</font> = <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><font color="Maroon">Id3</font>,<font color="Maroon">rid3</font>,<font color="Maroon">a3</font></span><a href="setwiseo.html#K4">.}</a></span>, <font color="Maroon">B2</font> = <span class="p1"><a href="setwiseo.html#K4">{.</a><span class="default"><font color="Maroon">a4</font>,<font color="Maroon">a5</font>,<font color="Maroon">a6</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#K12">Permutations</a> 3</span>)</span> ;<br/>
<a NAME="E30:61"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B2</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">B1</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</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">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B2</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</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">E86</font></i>: 
<font color="Maroon">B1</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_7_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 ;<br/><a NAME="E2:61_7_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Id3</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rid3</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font> )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="txt" href="matrix_9.html#E17:61"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E3:61_7_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, , </i>;<br/><b>hence </b><a NAME="E4:61_7_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B2</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">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B2</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">B1</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="txt" href="matrix_9.html#E16:61"><i><font color="Green">E72</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">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B2</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">E88</font></i>: the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B1</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</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">K</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Id3</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">rid3</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="setwop_2.html#T5">SETWOP_2:5</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r3</font>

;<br/>
<i><font color="Green">E141</font></i>: the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B2</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</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">K</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a4</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a5</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a6</font></span>)</span>
<b>by </b><i><a class="ref" href="setwop_2.html#T5">SETWOP_2:5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r4</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r5</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r6</font>

;<br/>
 <a href="matrix_3.html#K12">Det</a> <font color="Maroon">M</font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> 3</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D9">MATRIX_3:def 9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B1</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B2</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="setwop_2.html#T6">SETWOP_2:6</a>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r3</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r4</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">r5</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r6</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_9.html#E16:61"><i><font color="Green">E72</font></i></a>, <a class="txt" href="matrix_9.html#E17:61"><i><font color="Green">E73</font></i></a>, <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">r1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r3</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r4</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r5</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r6</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">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">e</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K10">*</a> <font color="Maroon">e</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">f</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">h</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">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">f</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">i</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</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#T3" target="_self">Th3</a>, <a class="ref" href="matrix_9.html#T5" target="_self">Th5</a>, , , , , <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E34:61"/>
 <a href="matrix_3.html#K12">Det</a> <font color="Maroon">M</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">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">e</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K10">*</a> <font color="Maroon">e</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">f</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">h</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">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">f</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">i</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</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">h</font></span>)</span>
 ;<br/>


</div>
