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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R1</font>, <font color="Olive" title="b2">R2</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="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Olive" title="b1">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Olive" title="b2">R2</font> holds <br/><span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Olive" title="b1">R1</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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Olive" title="b2">R2</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="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Olive" title="b2">R2</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>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R1</font>, <font color="Maroon" title="c3">R2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c2">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">R2</font> implies <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c2">R1</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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c3">R2</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="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c3">R2</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>) )</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">R1</font>, <font color="Olive" title="b2">R2</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="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Olive" title="b1">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Olive" title="b2">R2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Olive" title="b1">R1</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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Olive" title="b2">R2</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="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Olive" title="b2">R2</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/>
<span class="kw">assume </span><a NAME="E1:120"/><span class="lab"><font color="Green" title="E81">A1</font></span>: 
 <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c2">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c3">R2</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="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c2">R1</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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c3">R2</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="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c3">R2</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>)</span><br/>

<a NAME="E2:120"/><span class="lab"><font color="Green" title="E82">A2</font></span>: 
 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="120_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:120_1"/><span class="lab"><font color="Green" title="E83">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">n1</font> = <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">R1</font>, <font color="Maroon" title="c7">R2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c6">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c7">R2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c6">R1</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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c7">R2</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="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c7">R2</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>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:120_1"/><span class="lab"><font color="Green" title="E84">A4</font></span>: 
 <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c6">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c7">R2</font>
 <span class="kw">and </span><br/><a NAME="E3:120_1"/><span class="lab"><font color="Green" title="E85">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c6">R1</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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c7">R2</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="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c7">R2</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>)</span><br/>

<a NAME="E4:120_1"/><span class="lab"><font color="Green" title="E86">A6</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">R1n</font> = <font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font>;<br/>
<a NAME="E5:120_1"/><span class="lab"><font color="Green" title="E87">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:120_1"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">R2n</font> = <font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font>;<br/>
<span class="lab"><font color="Green" title="E88">A8</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">R1</font> = 
 <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="c7">R2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120_1"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">R2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>
;<br/>
<a NAME="E7:120_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:120_1"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">b2n</font> =  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">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/>
<span class="kw">set </span><font color="Maroon" title="c11">b1n</font> =  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">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/>
<span class="lab"><font color="Green" title="E90">A10</font></span>:  <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c6">R1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#T21" target="_self" title="MATRIXJ1:th.21">Th21</a></span>
<br/>.= 
 <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120_1"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="ref" href="matrixj1.html#T17" target="_self" title="MATRIXJ1:th.17">Th17</a></span>
;<br/>
<a NAME="E9:120_1"/><span class="lab"><font color="Green" title="E91">A11</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c6">R1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:120_1"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="ref" href="finseq_2.html#T124" title="FINSEQ_2:th.124">FINSEQ_2:124</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E92">A12</font></span>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c7">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120_1"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="txt" href="#E4:120_1"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="ref" href="matrixj1.html#D4" target="_self" title="MATRIXJ1:def.4">Def4</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120_1"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="txt" href="#E4:120_1"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="txt" href="#E9:120_1"><span class="lab"><font color="Green" title="E91">A11</font></span></a>, <a class="ref" href="matrixj1.html#D3" target="_self" title="MATRIXJ1:def.3">Def3</a></span>
;<br/>
<a NAME="E11:120_1"/><span class="lab"><font color="Green" title="E93">A13</font></span>: 
<font color="Maroon" title="c6">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">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="c6">R1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:120_1"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="ref" href="finseq_3.html#T55" title="FINSEQ_3:th.55">FINSEQ_3:55</a></span>;<br/>
<span class="lab"><font color="Green" title="E94">A14</font></span>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span>,<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="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#D5" target="_self" title="MATRIXJ1:def.5">Def5</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span>,<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="kw">by</span> <span class="lab"><a class="txt" href="#E8:120_1"><span class="lab"><font color="Green" title="E90">A10</font></span></a>, <a class="ref" href="matrixj1.html#D5" target="_self" title="MATRIXJ1:def.5">Def5</a></span>
;<br/>
<a NAME="E13:120_1"/><span class="lab"><font color="Green" title="E95">A15</font></span>: 
<font color="Maroon" title="c7">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">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="c7">R2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:120_1"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="txt" href="#E6:120_1"><span class="lab"><font color="Green" title="E88">A8</font></span></a>, <a class="ref" href="finseq_3.html#T55" title="FINSEQ_3:th.55">FINSEQ_3:55</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c6">R1</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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c7">R2</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="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</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="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">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="c6">R1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">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="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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</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="c7">R2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">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 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="kw">by</span> <span class="lab"><a class="txt" href="#E11:120_1"><span class="lab"><font color="Green" title="E93">A13</font></span></a>, <a class="ref" href="matrixj1.html#T35" target="_self" title="MATRIXJ1:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</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="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">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="c6">R1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">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="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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</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="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">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="c7">R2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">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="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="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#T35" target="_self" title="MATRIXJ1:th.35">Th35</a></span>
<br/>.= 
 <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><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="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">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="c6">R1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">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> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</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="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">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="c7">R2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:120_1"><span class="lab"><font color="Green" title="E94">A14</font></span></a>, <a class="txt" href="#E10:120_1"><span class="lab"><font color="Green" title="E92">A12</font></span></a>, <a class="ref" href="matrixj1.html#T78" target="_self" title="MATRIXJ1:th.78">Th78</a></span>
<br/>.= 
 <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1"><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">&lt;*</a><span class="default"><span class="p2">(<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="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">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> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <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="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">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>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">*&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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#T77" target="_self" title="MATRIXJ1:th.77">Th77</a></span>
<br/>.= 
 <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1"><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span></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"><span class="p3">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">*&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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:120_1"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="txt" href="#E5:120_1"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="txt" href="#E8:120_1"><span class="lab"><font color="Green" title="E90">A10</font></span></a></span>
<br/>.= 
 <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span></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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#T35" target="_self" title="MATRIXJ1:th.35">Th35</a></span>
<br/>.= 
 <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">R2</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2">(<span class="default"><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="c6">R1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">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> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</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="c7">R2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">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>,<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#T76" target="_self" title="MATRIXJ1:th.76">Th76</a></span>
<br/>.= 
 <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c7">R2</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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:120_1"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="txt" href="#E7:120_1"><span class="lab"><font color="Green" title="E89">A9</font></span></a>, <a class="txt" href="#E11:120_1"><span class="lab"><font color="Green" title="E93">A13</font></span></a>, <a class="txt" href="#E13:120_1"><span class="lab"><font color="Green" title="E95">A15</font></span></a>, <a class="ref" href="matrixj1.html#T74" target="_self" title="MATRIXJ1:th.74">Th74</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:120"/><span class="lab"><font color="Green" title="E83">A16</font></span>: 
<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="120_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">R1</font>, <font color="Maroon" title="c5">R2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c5">R2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">R1</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 <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c4">R1</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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c5">R2</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="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c5">R2</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>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:120_2"/><span class="lab"><font color="Green" title="E84">A17</font></span>: 
 <a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c5">R2</font>
 <span class="kw">and </span><br/><a NAME="E2:120_2"/><span class="lab"><font color="Green" title="E85">A18</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">R1</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c4">R1</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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c5">R2</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="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c5">R2</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>)</span><br/>

<a NAME="E3:120_2"/><span class="lab"><font color="Green" title="E86">A19</font></span>: 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c4">R1</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120_2"><span class="lab"><font color="Green" title="E85">A18</font></span></a>, <a class="ref" href="rvsum_1.html#T72" title="RVSUM_1:th.72">RVSUM_1:72</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">b2</font> =  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c5">R2</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>);<br/>
<a NAME="E4:120_2"/>
 <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c5">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:120_2"><span class="lab"><font color="Green" title="E84">A17</font></span></a>, <a class="txt" href="#E2:120_2"><span class="lab"><font color="Green" title="E85">A18</font></span></a></span>;<br/>
<a NAME="E5:120_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A20</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c5">R2</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="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#D5" target="_self" title="MATRIXJ1:def.5">Def5</a>, <a class="ref" href="rvsum_1.html#T72" title="RVSUM_1:th.72">RVSUM_1:72</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">b1</font> =  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c4">R1</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>);<br/>
<a NAME="E6:120_2"/><span class="lab"><font color="Green" title="E88">A21</font></span>: 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <font color="Maroon" title="c4">R1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c4">R1</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="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#D5" target="_self" title="MATRIXJ1:def.5">Def5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">b12</font> =  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c5">R2</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/>
<a NAME="E7:120_2"/><span class="lab"><font color="Green" title="E89">A22</font></span>: 
 <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120_2"><span class="lab"><font color="Green" title="E85">A18</font></span></a></span>;<br/>
<a NAME="E8:120_2"/>
 <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c5">R2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:120_2"><span class="lab"><font color="Green" title="E84">A17</font></span></a>, <a class="ref" href="matrixj1.html#T73" target="_self" title="MATRIXJ1:th.73">Th73</a></span>;<br/>
<a NAME="E9:120_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c5">R2</font></span>)</span>,<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="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:120_2"><span class="lab"><font color="Green" title="E89">A22</font></span></a>, <a class="ref" href="matrixj1.html#D5" target="_self" title="MATRIXJ1:def.5">Def5</a>, <a class="ref" href="rvsum_1.html#T72" title="RVSUM_1:th.72">RVSUM_1:72</a></span>;<br/>
<a NAME="E10:120_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A23</font></span>: 
 <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c5">R2</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>) <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="E11:120_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c4">R1</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="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:120_2"><span class="lab"><font color="Green" title="E89">A22</font></span></a>, <a class="ref" href="matrixj1.html#D5" target="_self" title="MATRIXJ1:def.5">Def5</a>, <a class="ref" href="rvsum_1.html#T72" title="RVSUM_1:th.72">RVSUM_1:72</a></span>;<br/>
<a NAME="E12:120_2"/><span class="kw">then </span>
 <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="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c4">R1</font>,<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> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c5">R2</font>,<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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:120_2"><span class="lab"><font color="Green" title="E87">A20</font></span></a>, <a class="txt" href="#E6:120_2"><span class="lab"><font color="Green" title="E88">A21</font></span></a>, <a class="txt" href="#E3:120_2"><span class="lab"><font color="Green" title="E86">A19</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:120_2"/>
<span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c4">R1</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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c5">R2</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="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c5">R2</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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:120_2"><span class="lab"><font color="Green" title="E90">A23</font></span></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/>


</div><span class="kw">end;</span></div>
<a NAME="E4:120"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E3:120"><span class="lab"><font color="Green" title="E83">A16</font></span></a>, <a class="txt" href="#E2:120"><span class="lab"><font color="Green" title="E82">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:120"/>
<span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c2">R1</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="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> (<font color="Maroon" title="c3">R2</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="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K26" title="MATRIXJ1:func.26">(#)</a> <font color="Maroon" title="c3">R2</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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:120"><span class="lab"><font color="Green" title="E81">A1</font></span></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/>


</div>
