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<div class="add">

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">K</font>;<br/>



<a NAME="E1:45"/><i><font color="Green" title="E38">A1</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c3">p</font> =  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <b>by </b><i><a class="txt" href="matrixr2.html#E44"><i><font color="Green" title="E35">Lm4</font></i></a>, <a class="ref" href="matrix_2.html#D11" title="MATRIX_2:def.11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E2:45_1"/><i><font color="Green" title="E39">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E3:45_1"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">K</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T32" title="GROUP_1:th.32">GROUP_1:32</a></i>;<br/>
<a NAME="E4:45_1"/><b>then </b><i><font color="Green" title="E40">A3</font></i>: 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 
<b>by </b><i><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></i>;<br/>
<i><font color="Green" title="E41">A4</font></i>: the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">A</font></span>)</span> = 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">K</font>
<b>by </b><i><a class="txt" href="matrixr2.html#E2:45_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="matrixr2.html#E4:45_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="finsop_1.html#D1" title="FINSOP_1:def.1">FINSOP_1:def 1</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">K</font>
<b>by </b><i><a class="ref" href="group_1.html#T33" title="GROUP_1:th.33">GROUP_1:33</a></i>
;<br/>
<a NAME="E6:45_1"/>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">K</font></span>)</span>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:45_1_2"/><i><font color="Green" title="E42">A5</font></i>: 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#D1" title="FINSEQ_2:def.1">FINSEQ_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">q</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> as    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <b>by </b><i><a class="txt" href="matrixr2.html#E44"><i><font color="Green" title="E35">Lm4</font></i></a>, <a class="ref" href="matrix_2.html#D11" title="MATRIX_2:def.11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E3:45_1_2"/>
 <a href="matrix_2.html#K11" title="MATRIX_2:func.11">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="matrix_2.html#T20" title="MATRIX_2:th.20">MATRIX_2:20</a></i>;<br/>
<a NAME="E4:45_1_2"/><b>then </b>
<font color="Maroon" title="c4">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T29" title="MATRIX_2:th.29">MATRIX_2:29</a></i>;<br/>
<b>hence </b><a NAME="E5:45_1_2"/>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">K</font></span>)</span>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">K</font>
 <b>by </b><i><a class="txt" href="matrixr2.html#E1:45_1_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="matrix_2.html#D16" title="MATRIX_2:def.16">MATRIX_2:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:45_1"/>
<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font>
 <b>by </b><i><a class="txt" href="matrixr2.html#E5:45_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="matrix_3.html#D8" title="MATRIX_3:def.8">MATRIX_3:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c3">kk</font> =  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <b>by </b><i><a class="ref" href="matrixr2.html#T40" target="_self" title="MATRIXR2:th.40">Th40</a>, <a class="ref" href="finseq_2.html#T58" title="FINSEQ_2:th.58">FINSEQ_2:58</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green" title="E39">A6</font></i>:  <a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
 <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="matrixr2.html#T40" target="_self" title="MATRIXR2:th.40">Th40</a>, <a class="ref" href="matrix_2.html#D17" title="MATRIX_2:def.17">MATRIX_2:def 17</a></i>
<br/>.= 
<span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c3">kk</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>
<b>by </b><i><a class="txt" href="matrixr2.html#E44"><i><font color="Green" title="E35">Lm4</font></i></a>, <a class="ref" href="matrixr2.html#T39" target="_self" title="MATRIXR2:th.39">Th39</a>, <a class="ref" href="matrixr2.html#T40" target="_self" title="MATRIXR2:th.40">Th40</a></i>
;<br/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c2">A</font> = 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c2">A</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D9" title="MATRIX_3:def.9">MATRIX_3:def 9</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font>
<b>by </b><i><a class="txt" href="matrixr2.html#E1:45"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="matrixr2.html#E3:45"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></i>
;<br/>
<b>hence </b><a NAME="E5:45"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font>
 ;<br/>


</div>
