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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <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> holds   <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Olive" title="b1">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></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">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#M1" title="MATRIX_2:mode.1">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T40" target="_self" title="MATRIXR2:th.40">Th40</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">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></span><br/>

<a NAME="E2:44"/><span class="lab"><font color="Green" title="E37">A1</font></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="c3">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="44_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c4">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#M1" title="MATRIX_2:mode.1">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="matrixr2.html#E44"><span class="lab"><font color="Green" title="E34">Lm4</font></span></a>, <a class="ref" href="matrix_2.html#D9" title="MATRIX_2:def.9">MATRIX_2:def 9</a></span>;<br/>
<a NAME="E2:44_1"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span>,<font color="Maroon" title="c4">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">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#M1" title="MATRIX_2:mode.1">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="matrixr2.html#E44"><span class="lab"><font color="Green" title="E34">Lm4</font></span></a>, <a class="ref" href="matrix_2.html#D9" title="MATRIX_2:def.9">MATRIX_2:def 9</a></span>;<br/>
<a NAME="E2:44_1_1"/>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>;<br/>
<a NAME="E3:44_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T25" title="MATRIX_2:th.25">MATRIX_2:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:44_1_1"/>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span>,<font color="Maroon" title="c4">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:44_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T21" title="GROUP_1:th.21">GROUP_1:21</a></span>;<br/>
<a NAME="E4:44_1"/><span class="kw">then </span>
(  <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="c4">p</font>,<font color="Maroon" title="c3">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>  &amp;  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a>, <a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></span>;<br/>
<span class="kw">then </span> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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="c4">p</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> = 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#D1" title="FINSOP_1:def.1">FINSOP_1:def 1</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T22" title="GROUP_1:th.22">GROUP_1:22</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:44_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="c3">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="matrix_3.html#D8" title="MATRIX_3:def.8">MATRIX_3:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E38">A3</font></span>:  <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T40" target="_self" title="MATRIXR2:th.40">Th40</a>, <a class="ref" href="matrix_2.html#D14" title="MATRIX_2:def.14">MATRIX_2:def 14</a></span>
<br/>.= 
<span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c2">kk</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T40" target="_self" title="MATRIXR2:th.40">Th40</a></span>
;<br/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">A</font> = 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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="c3">A</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D9" title="MATRIX_3:def.9">MATRIX_3:def 9</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="setwiseo.html#T17" title="SETWISEO:th.17">SETWISEO:17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:44"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
