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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/>  for <font color="Olive" title="b2">M</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">K</font>  st <font color="Maroon" title="c1">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="c2">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b2">M</font>,<font color="Maroon" title="c3">j</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"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Olive" title="b2">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b2">M</font>,<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b2">M</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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">M</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c4">K</font>  st <font color="Maroon" title="c1">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="c2">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c3">j</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"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b1">M</font>,<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">M</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c4">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">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="c2">n</font> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">j</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"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">M</font>,<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span> )</span><br/>









<span class="kw">assume </span><a NAME="E1:39"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c1">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="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">j</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"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">M</font>,<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">C</font> =  <a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c5">M</font>;<br/>
<a NAME="E2:39"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E3:39"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p1">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c5">M</font></span>)</span>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c5">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>,<font color="Maroon" title="c1">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="matrix_2.html#T14" title="MATRIX_2:th.14">MATRIX_2:14</a></span>;<br/>
<a NAME="E5:39"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E6:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<span class="kw">thus </span> <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="c5">M</font>,<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span> = 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> (<span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> (<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c1">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="laplace.html#T25" target="_self" title="LAPLACE:th.25">Th25</a></span>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</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="c5">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>,<font color="Maroon" title="c1">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">j</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="laplace.html#T28" target="_self" title="LAPLACE:th.28">Th28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">j</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"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>,<font color="Maroon" title="c1">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T64" title="FVSUM_1:th.64">FVSUM_1:64</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/>


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