<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>;<br/>





<b>set </b><font color="Maroon" title="c4">D</font> =  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>set </b><font color="Maroon" title="c5">D'</font> = <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">C</font> =  <a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>set </b><font color="Maroon" title="c7">DC</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">MC</font> = <font color="Maroon" title="c3">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">ID</font> =  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>;<br/>
<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E33">A1</font></i>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E34">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c12">i'</font> = <font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c13">j'</font> = <font color="Maroon" title="c11">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:40_1"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E4:40_1"/><b>then </b><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="laplace.html#E1:40_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">N</font> = <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">COL</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c11">j</font>, <font color="Maroon" title="c16">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">N</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E7:40_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j</font> &amp; <font color="Maroon" title="c11">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="laplace.html#E4:40_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:40_1"/><b>then </b>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <font color="Maroon" title="c15">COL</font>
 <b>by </b><i><a class="ref" href="matrixr1.html#T19" title="MATRIXR1:th.19">MATRIXR1:19</a></i>;<br/><a NAME="E9:40_1"/><b>then </b>
 <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span>,<font color="Maroon" title="c11">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12" title="FVSUM_1:func.12">mlt</a> <font color="Maroon" title="c16">L</font>,<font color="Maroon" title="c15">COL</font></span>)</span>
 <b>by </b><i><a class="ref" href="fvsum_1.html#T83" title="FVSUM_1:th.83">FVSUM_1:83</a></i>;<br/><b>then </b><i><font color="Green" title="E36">A4</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span>,<font color="Maroon" title="c11">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c11">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T92" title="FVSUM_1:th.92">FVSUM_1:92</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c13">j'</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c12">i'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="laplace.html#E4:40_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="laplace.html#T29" target="_self" title="LAPLACE:th.29">Th29</a></i>
;<br/><a NAME="E11:40_1"/><b>then </b><i><font color="Green" title="E37">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">j</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="laplace.html#E1:40_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="laplace.html#E4:40_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/><a NAME="E12:40_1"/><i><font color="Green" title="E38">A6</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_2"/>
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font> or <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1"><b>suppose </b></a><a NAME="E1:40_1_2_1"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font>
 ;<br/><div class="add"><a NAME="E2:40_1_2_1"/><b>then </b>
(  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c11">j</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</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="c2">K</font> &amp; <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <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="c2">K</font> )
 <b>by </b><i><a class="txt" href="laplace.html#E1:40"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="laplace.html#E1:40_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="laplace.html#E12:40_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a>, <a class="ref" href="matrix_1.html#D12" title="MATRIX_1:def.12">MATRIX_1:def 12</a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a></i>;<br/><b>hence </b><a NAME="E3:40_1_2_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>
 <b>by </b><i><a class="txt" href="laplace.html#E1:40_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="laplace.html#E4:40_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="laplace.html#E10:40_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_2"><b>suppose </b></a><a NAME="E1:40_1_2_2"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">j</font>
 ;<br/><div class="add"><a NAME="E2:40_1_2_2"/><b>then </b>
(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c13">j'</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c12">i'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="laplace.html#E1:40_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="laplace.html#E4:40_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="laplace.html#E12:40_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="matrix11.html#T51" title="MATRIX11:th.51">MATRIX11:51</a>, <a class="ref" href="matrix_1.html#D12" title="MATRIX_1:def.12">MATRIX_1:def 12</a></i>;<br/><b>hence </b><a NAME="E3:40_1_2_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>
 <b>by </b><i><a class="txt" href="laplace.html#E11:40_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
<font color="Maroon" title="c3">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></i>;<br/>


</div>
