<?xml version="1.0"?>
<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">R</font> being  <a href="matrixj1.html#NM4" title="MATRIXJ1:NM.4">FinSequence_of_Square-Matrix</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"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b1">R</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Olive" title="b1">R</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="matrixj1.html#NM4" title="MATRIXJ1:NM.4">FinSequence_of_Square-Matrix</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"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c2">R</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R</font></span>)</span></span><br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">R</font> being  <a href="matrixj1.html#NM4" title="MATRIXJ1:NM.4">FinSequence_of_Square-Matrix</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b1">R</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Olive" title="b1">R</font></span>)</span>;<br/>
<a NAME="E1:85"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">R</font> be   <a href="matrixj1.html#NM4" title="MATRIXJ1:NM.4">FinSequence_of_Square-Matrix</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c3">R</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:85_1"/><i><font color="Green" title="E57">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <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"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c3">R</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R</font></span>)</span></span><br/>

<a NAME="E2:85_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:85_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:85_1"/><b>then </b>
 <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E4:85_1"/><b>then </b>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></i>;<br/>
<b>hence </b> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c3">R</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font>
<b>by </b><i><a class="ref" href="matrixr2.html#T41" title="MATRIXR2:th.41">MATRIXR2:41</a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:85_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="fvsum_1.html#T105" title="FVSUM_1:th.105">FVSUM_1:105</a></i>
;<br/> <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>
<a NAME="E2:85"/><i><font color="Green" title="E57">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:85_2"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<b>set </b><font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>let </b><font color="Maroon" title="c5">R</font> be   <a href="matrixj1.html#NM4" title="MATRIXJ1:NM.4">FinSequence_of_Square-Matrix</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c5">R</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c5">R</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:85_2"/><i><font color="Green" title="E59">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c5">R</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c5">R</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c6">Rn</font> = <font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font>;<br/>
<b>set </b><font color="Maroon" title="c7">bR</font> =  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">R1</font> = <font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<a NAME="E3:85_2"/><i><font color="Green" title="E60">A6</font></i>: 
<font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p1"><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E2:85_2"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="finseq_3.html#T61" title="FINSEQ_3:th.61">FINSEQ_3:61</a></i>;<br/>
<i><font color="Green" title="E61">A7</font></i>:  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p2"><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixj1.html#T16" target="_self" title="MATRIXJ1:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixj1.html#D5" target="_self" title="MATRIXJ1:def.5">Def5</a></i>
<br/>.= 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
<br/>.= 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p3"><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixj1.html#T15" target="_self" title="MATRIXJ1:th.15">Th15</a></i>
<br/>.= 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c5">R</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:85_2"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="matrixj1.html#T14" target="_self" title="MATRIXJ1:th.14">Th14</a></i>
;<br/>
<a NAME="E5:85_2"/><i><font color="Green" title="E62">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="#E2:85_2"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E6:85_2"/>
 <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c5">R</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <span class="p1"><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="matrixj1.html#K17" title="MATRIXJ1:func.17">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:85_2"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="matrixj1.html#T35" target="_self" title="MATRIXJ1:th.35">Th35</a></i>;<br/>
<b>hence </b> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c5">R</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> = 
<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="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:85_2"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="matrixj1.html#T52" target="_self" title="MATRIXJ1:th.52">Th52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:85_2"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="#E5:85_2"><i><font color="Green" title="E62">A8</font></i></a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T9" title="GROUP_4:th.9">GROUP_4:9</a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p3"><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixj1.html#T49" target="_self" title="MATRIXJ1:th.49">Th49</a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c5">R</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:85_2"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="matrixj1.html#T50" target="_self" title="MATRIXJ1:th.50">Th50</a></i>
;<br/> <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>
<a NAME="E3:85"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E1:85"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="#E2:85"><i><font color="Green" title="E57">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:85"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c2">R</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R</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>
