<?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">A</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>assume </b><a NAME="E1:52"/><i><font color="Green" title="E40">A1</font></i>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">A</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/>

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font>;<br/>

<b>assume </b><a NAME="E2:52"/><i><font color="Green" title="E41">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">x</font> <a href="laplace.html#K12" title="LAPLACE:func.12">*</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">X</font> = <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c7">B</font> = <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span>;<br/>
<a NAME="E3:52"/><i><font color="Green" title="E42">A3</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/>
<b>hence </b><a NAME="E4:52"/>
<span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="laplace.html#K12" title="LAPLACE:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="laplace.html#E1:52"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="laplace.html#E2:52"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="laplace.html#T38" target="_self" title="LAPLACE:th.38">Th38</a></i>;<br/>

<b>let </b><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E5:52"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c8">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>
 ;<br/>

<a NAME="E6:52"/>
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> 1
 
;<br/>
<a NAME="E7:52"/><b>then </b><i><font color="Green" title="E44">A5</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c8">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 1</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> &amp; <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> 1</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> <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"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 
<b>by </b><i><a class="txt" href="laplace.html#E2:52"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="laplace.html#E5:52"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c8">i</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="laplace.html#E2:52"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="laplace.html#E3:52"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="laplace.html#E5:52"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="laplace.html#E7:52"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/>
<b>hence </b><font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</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">A</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c8">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span>,1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="laplace.html#E1:52"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="laplace.html#E2:52"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="laplace.html#E3:52"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="laplace.html#E7:52"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="laplace.html#T38" target="_self" title="LAPLACE:th.38">Th38</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">A</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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#K3" title="MATRIX11:func.3">ReplaceLine</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="laplace.html#E7:52"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/><br/>


</div>
