<?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>; <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">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="c1">n</font>,<font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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> holds <br/> <a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Olive" title="b3">i</font></span><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>; <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">M</font> being  <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/> for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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> holds <br/> <a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Olive" title="b2">i</font></span><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>; <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">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> holds <br/> <a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Olive" title="b1">i</font></span><br/><b>let </b><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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> implies  <a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font> )</span><br/>







<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c4">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">I</font> = <font color="Maroon" title="c4">i</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>set </b><font color="Maroon" title="c6">LC</font> =  <a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>;<br/>
<b>set </b><font color="Maroon" title="c7">LL</font> =  <a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font>;<br/>
<a NAME="E3:36"/><i><font color="Green" title="E31">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></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"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></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="ref" href="laplace.html#D7" target="_self" title="LAPLACE:def.7">Def7</a>, <a class="ref" href="laplace.html#D8" target="_self" title="LAPLACE:def.8">Def8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )</span><br/><b>assume </b><a NAME="E1:36_1"/><i><font color="Green" title="E32">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/><a NAME="E2:36_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="E3:36_1"/><b>then </b>
( <font color="Maroon" title="c8">k</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#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font> <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="txt" href="#E1:36_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E4:36_1"/><b>then </b>
( <font color="Maroon" title="c8">k</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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</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="c1">n</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</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="c1">n</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c4">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>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font> )
 <b>by </b><i><a class="txt" href="#E1:36"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="#E3:36"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:36_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</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_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c5">I</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c5">I</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c8">k</font></span>)</span> &amp;  <a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c5">I</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_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="#E1:36"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="laplace.html#D7" target="_self" title="LAPLACE:def.7">Def7</a>, <a class="ref" href="laplace.html#D8" target="_self" title="LAPLACE:def.8">Def8</a>, <a class="ref" href="laplace.html#T24" target="_self" title="LAPLACE:th.24">Th24</a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>;<br/><b>hence </b><a NAME="E6:36_1"/>
<span class="p1">(<span class="default"><a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:36"/>
 <a href="laplace.html#K9" title="LAPLACE:func.9">LaplaceExpC</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K8" title="LAPLACE:func.8">LaplaceExpL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="#E3:36"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
