<?xml version="1.0"?>
<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> 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>3</sub></font> = <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span>;<br/>
<a NAME="E1:47"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11" target="_self">Path_product</a> <span class="p2"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1">idseq</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="finseq_2.html#K1">idseq</a> 1 as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 1 <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E2:47_1"/><i><font color="Green">E25</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" target="_self">Path_matrix</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:47_1"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" target="_self">Path_matrix</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:47_1"/><b>then </b><i><font color="Green">E27</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" target="_self">Path_matrix</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:47_1"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="matrix_3.html#E3:47_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E6:47_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_3.html#K10" target="_self">Path_matrix</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> <a href="matrix_1.html#K3">*</a> 1,1
 
<b>by </b><i><a class="txt" href="matrix_3.html#E4:47_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="matrix_3.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E7:47_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_3.html#K10" target="_self">Path_matrix</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span></span>)</span> <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="matrix_2.html#T5">MATRIX_2:5</a></i>;<br/>
<a NAME="E8:47_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="matrix_3.html#K10" target="_self">Path_matrix</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E2:47_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E9:47_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11" target="_self">Path_product</a> <span class="p2"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <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">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" target="_self">Path_matrix</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p4"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E10:47_1"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T73">FINSEQ_2:73</a></i>;<br/>
<a NAME="E11:47_1"/>
 <a href="matrix_2.html#K14">-</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 1</span>)</span> as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 1 <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E2:47_1_1"/>
 <a href="matrix_2.html#K10">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrix_2.html#T20">MATRIX_2:20</a></i>;<br/>
<a NAME="E3:47_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> 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/>
<b>hence </b><a NAME="E4:47_1_1"/>
 <a href="matrix_2.html#K14">-</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:47_1"/>
<span class="p1">(<span class="default"><a href="matrix_3.html#K11" target="_self">Path_product</a> <span class="p2"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1">idseq</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="matrix_3.html#E8:47_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="matrix_3.html#E9:47_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="matrix_3.html#E10:47_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finsop_1.html#T17">FINSOP_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47"/><i><font color="Green">E25</font></i>: 
 <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_2.html#K1">idseq</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T21">MATRIX_2:21</a>, <a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<a NAME="E3:47"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_2.html#K1">idseq</a> 1 <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K11">Permutations</a> 1
 
<b>by </b><i><a class="ref" href="matrix_2.html#T21">MATRIX_2:21</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:47"/>
 <a href="matrix_3.html#K12" target="_self">Det</a> <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E1:47"><i><font color="Green">E24</font></i></a>, <a class="txt" href="matrix_3.html#E2:47"><i><font color="Green">E25</font></i></a>, <a class="txt" href="matrix_3.html#E3:47"><i><font color="Green">E26</font></i></a>, <a class="ref" href="setwiseo.html#T26">SETWISEO:26</a></i>;<br/>
<b>hence </b><a NAME="E5:47"/>
 <a href="matrix_3.html#K12" target="_self">Det</a> <span class="p1"><a href="matrix_2.html#K5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span></span><a href="matrix_2.html#K5">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
