<?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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">a1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">G</font> being  <a href="matrixj1.html#NM2" title="MATRIXJ1:NM.2">FinSequence_of_Matrix</a> of <font color="Maroon" title="c1">K</font> holds  <font color="Olive" title="b1">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Olive" title="b3">G</font>,<font color="Olive" title="b2">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b3">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">a1</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">a1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="matrixj1.html#NM2" title="MATRIXJ1:NM.2">FinSequence_of_Matrix</a> of <font color="Maroon" title="c1">K</font> holds  <font color="Maroon" title="c2">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c3">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Olive" title="b1">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">a1</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">G</font> be   <a href="matrixj1.html#NM2" title="MATRIXJ1:NM.2">FinSequence_of_Matrix</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c3">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c4">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">a1</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">a</font>, <font color="Olive" title="b2">a1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">G</font> being  <a href="matrixj1.html#NM2" title="MATRIXJ1:NM.2">FinSequence_of_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="b3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Olive" title="b1">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Olive" title="b3">G</font>,<font color="Olive" title="b2">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b3">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">a1</font></span>)</span>;<br/>
<a NAME="E1:101"/><span class="lab"><font color="Green" title="E65">A1</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="101_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">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="c6">n1</font> = <font color="Maroon" title="c5">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="c7">a</font>, <font color="Maroon" title="c8">a1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="matrixj1.html#NM2" title="MATRIXJ1:NM.2">FinSequence_of_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">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c8">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c7">a</font></span>)</span>,<font color="Olive" title="b1">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">a1</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">G</font> be   <a href="matrixj1.html#NM2" title="MATRIXJ1:NM.2">FinSequence_of_Matrix</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c9">G</font>,<font color="Maroon" title="c8">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c7">a</font></span>)</span>,<font color="Maroon" title="c9">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">a1</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E2:101_1"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c9">G</font>,<font color="Maroon" title="c8">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c7">a</font></span>)</span>,<font color="Maroon" title="c9">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">a1</font></span>)</span></span><br/>

<a NAME="E3:101_1"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K6" title="MATRIXJ1:func.6">|</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p1"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="finseq_3.html#T61" title="FINSEQ_3:th.61">FINSEQ_3:61</a></span>;<br/>
<a NAME="E4:101_1"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:101_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A5</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="c9">G</font> <a href="matrixj1.html#K6" title="MATRIXJ1:func.6">|</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<a NAME="E6:101_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
;<br/>
<a NAME="E7:101_1"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<a NAME="E8:101_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A6</font></span>: 
<span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E9:101_1"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
<span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<a NAME="E10:101_1"/><span class="lab"><font color="Green" title="E72">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/>
<a NAME="E11:101_1"/><span class="lab"><font color="Green" title="E73">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/>
<a NAME="E12:101_1"/><span class="lab"><font color="Green" title="E74">A10</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="c5">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E13:101_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T74" title="FINSEQ_2:th.74">FINSEQ_2:74</a></span>;<br/>
<span class="kw">hence </span> <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c7">a</font></span>)</span>,<font color="Maroon" title="c9">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">a1</font></span>)</span> = 
 <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K6" title="MATRIXJ1:func.6">|</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="matrixj1.html#K3" title="MATRIXJ1:func.3">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>,<span class="p3"><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E3:101_1"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E5:101_1"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E12:101_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E11:101_1"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="txt" href="#E10:101_1"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="ref" href="matrixj1.html#T64" target="_self" title="MATRIXJ1:th.64">Th64</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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K6" title="MATRIXJ1:func.6">|</a> <font color="Maroon" title="c5">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"><font color="Maroon" title="c7">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">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"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:101_1"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="txt" href="#E9:101_1"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="matrixj1.html#T63" target="_self" title="MATRIXJ1:th.63">Th63</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#K4" title="MATRIXJ1:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p4">(<span class="default"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c7">a</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K6" title="MATRIXJ1:func.6">|</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">a1</font></span>)</span></span>)</span></span><a href="matrixj1.html#K4" title="MATRIXJ1:func.4">*&gt;</a></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"><font color="Maroon" title="c7">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">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"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">a1</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"><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K6" title="MATRIXJ1:func.6">|</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c8">a1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">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"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E5:101_1"><span class="lab"><font color="Green" title="E69">A5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p2"><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K6" title="MATRIXJ1:func.6">|</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c8">a1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="matrixj1.html#K2" title="MATRIXJ1:func.2">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">*&gt;</a></span>,<font color="Maroon" title="c8">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="matrixj1.html#E80"><span class="lab"><font color="Green" title="E64">Lm7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c9">G</font>,<font color="Maroon" title="c8">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="matrixj1.html#T35" target="_self" title="MATRIXJ1:th.35">Th35</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">M</font> =  <a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c7">a</font></span>)</span>,<font color="Maroon" title="c9">G</font>;<br/>

</div><span class="kw">end;</span></div>
<a NAME="E2:101"/><span class="lab"><font color="Green" title="E66">A11</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="101_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">a1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="matrixj1.html#NM2" title="MATRIXJ1:NM.2">FinSequence_of_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">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  holds <br/><font color="Maroon" title="c5">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c6">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Olive" title="b1">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">a1</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">G</font> be   <a href="matrixj1.html#NM2" title="MATRIXJ1:NM.2">FinSequence_of_Matrix</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</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 <font color="Maroon" title="c5">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c6">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c7">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">a1</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:101_2"/><span class="lab"><font color="Green" title="E67">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c6">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c7">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">a1</font></span>)</span></span><br/>

<a NAME="E2:101_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#D9" target="_self" title="MATRIXJ1:def.9">Def9</a></span>;<br/>
<a NAME="E3:101_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#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E4:101_2"/><span class="kw">then </span>
 <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c7">G</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_2"><span class="lab"><font color="Green" title="E67">A12</font></span></a></span>;<br/>
<a NAME="E5:101_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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c7">G</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">a1</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#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></span>;<br/>
<a NAME="E6:101_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A13</font></span>: 
 <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c7">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">a1</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="E7:101_2"/>
 <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c7">G</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="#E1:101_2"><span class="lab"><font color="Green" title="E67">A12</font></span></a></span>;<br/>
<a NAME="E8:101_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> <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c6">a1</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="ref" href="matrixj1.html#D5" target="_self" title="MATRIXJ1:def.5">Def5</a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></span>;<br/>
<a NAME="E9:101_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"><font color="Maroon" title="c5">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c6">a1</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="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:101_2"/>
<font color="Maroon" title="c5">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c6">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c7">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">a1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_2"><span class="lab"><font color="Green" title="E68">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:101"/>
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="#E2:101"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E65">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:101"/>
<font color="Maroon" title="c2">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c3">a1</font></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"><a href="matrixj1.html#K24" title="MATRIXJ1:func.24">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c4">G</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">a1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
