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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> holds   <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Olive" title="b4">d</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Olive" title="b3">c</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font></span>)</span></span><br/>









<b>set </b><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">id2</font> =  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 2 as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">Id2</font> =  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 2 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> 2 <b>by </b><i><a class="ref" href="matrix_2.html#D11" title="MATRIX_2:def.11">MATRIX_2:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">rid2</font> =  <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 2</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> 2 <b>by </b><i><a class="ref" href="matrix_9.html#T4" target="_self" title="MATRIX_9:th.4">Th4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">F</font> = the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font>;<br/>
<b>set </b><font color="Maroon" title="c11">f</font> =  <a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span>;<br/>
<a NAME="E4:22"/>
 <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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<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="E5:22"/><b>then </b><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c8">Id2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">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/>
<i><font color="Green" title="E16">A2</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">id2</font> = 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default">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">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c8">Id2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c8">Id2</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D8" title="MATRIX_3:def.8">MATRIX_3:def 8</a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</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="c8">Id2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_9.html#E5:22"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="matrix_2.html#D16" title="MATRIX_2:def.16">MATRIX_2:def 16</a></i>
<br/>.= 
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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
<b>by </b><i><a class="ref" href="matrix_9.html#T9" target="_self" title="MATRIX_9:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font>
<b>by </b><i><a class="ref" href="matrix_9.html#T11" target="_self" title="MATRIX_9:th.11">Th11</a></i>
;<br/>
<a NAME="E7:22"/><i><font color="Green" title="E17">A3</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
;<br/>
<a NAME="E8:22"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c9">rid2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">2,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T61" title="FINSEQ_2:th.61">FINSEQ_2:61</a>, <a class="ref" href="finseq_5.html#T64" title="FINSEQ_5:th.64">FINSEQ_5:64</a></i>;<br/>
<a NAME="E9:22"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c7">id2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">rid2</font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E7:22"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="matrix_9.html#T2" target="_self" title="MATRIX_9:th.2">Th2</a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<a NAME="E10:22"/><i><font color="Green" title="E20">A6</font></i>: 
 <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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="matrix_2.html#T20" title="MATRIX_2:th.20">MATRIX_2:20</a></i>;<br/>
<i><font color="Green" title="E21">A7</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">rid2</font> = 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default">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">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c9">rid2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c9">rid2</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D8" title="MATRIX_3:def.8">MATRIX_3:def 8</a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c9">rid2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_9.html#E8:22"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="matrix_9.html#E10:22"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="matrix_9.html#T12" target="_self" title="MATRIX_9:th.12">Th12</a>, <a class="ref" href="matrix_2.html#D16" title="MATRIX_2:def.16">MATRIX_2:def 16</a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default">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="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#T10" target="_self" title="MATRIX_9:th.10">Th10</a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#T11" target="_self" title="MATRIX_9:th.11">Th11</a></i>
;<br/>
<a NAME="E12:22"/>
 <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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> 2
 
<b>by </b><i><a class="ref" href="matrix_2.html#T30" title="MATRIX_2:th.30">MATRIX_2:30</a>, <a class="ref" href="matrix_2.html#D17" title="MATRIX_2:def.17">MATRIX_2:def 17</a></i>;<br/>
<b>then </b> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span> = 
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"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Maroon" title="c8">Id2</font>,<font color="Maroon" title="c9">rid2</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#T6" target="_self" title="MATRIX_9:th.6">Th6</a>, <a class="ref" href="matrix_3.html#D9" title="MATRIX_3:def.9">MATRIX_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_9.html#E6:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="matrix_9.html#E9:22"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="matrix_9.html#E11:22"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="setwop_2.html#T3" title="SETWOP_2:th.3">SETWOP_2:3</a></i>
;<br/>
<b>hence </b><a NAME="E14:22"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
