<?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">V1</font> being  <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">V1</font> holds   <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Olive" title="b2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V1</font> be   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</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">M</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font> holds   <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></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">M</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span><br/>

<a NAME="E1:49"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V1</font> holds   <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V1</font>] <span class="kw">means </span> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default">$1</span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default">$1</span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">P</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span><br/>

<a NAME="E1:49_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&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="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></span>;<br/>
<a NAME="E2:49_1"/><span class="kw">then </span>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></span>;<br/>
<a NAME="E3:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<a NAME="E4:49_1"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V1</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V1</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">P</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">P</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V1</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">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V1</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">P</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V1</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>2</sub></font>[<font color="Maroon" title="c5">P</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>] )</span><br/>



<span class="kw">assume </span><a NAME="E1:49_1_1"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 ; <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>2</sub></font>[<font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>]</span><br/>

 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span></span>)</span> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D4" title="PRE_POLY:def.4">PRE_POLY:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1

;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E40">A5</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> = 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span>
<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="E4:49_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E42">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">i</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" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:49_1_1_3"/><span class="lab"><font color="Green" title="E43">A8</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/><a NAME="E2:49_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><a NAME="E3:49_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:49_1_1_3"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="txt" href="#E1:49_1_1_3"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><a NAME="E5:49_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">m1</font> = <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c9">m2</font> = <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D3" title="PRE_POLY:def.3">PRE_POLY:def 3</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
<font color="Maroon" title="c8">m1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c9">m2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_1_3"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="pre_poly.html#D4" title="PRE_POLY:def.4">PRE_POLY:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c9">m2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_1_3"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_1_3"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>
<br/>.= 
<span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_1_3"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</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>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_4"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:49_1_1_4"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_4_1"><span class="kw">suppose </span></a><a NAME="E1:49_1_1_4_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">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">S<sub>2</sub></font>[<font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>]</span><br/><div class="add"><a NAME="E2:49_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A10</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><span class="kw">hence </span> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> = 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T19" target="_self" title="MATRLIN:th.19">Th19</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p5"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1_1_4_1"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_4_2"><span class="kw">suppose </span></a><a NAME="E1:49_1_1_4_2"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">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">S<sub>2</sub></font>[<font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>]</span><br/><div class="add"><a NAME="E2:49_1_1_4_2"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&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#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A13</font></span>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>
;<br/><a NAME="E4:49_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E5:49_1_1_4_2"/><span class="lab"><font color="Green" title="E47">A15</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&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#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><a NAME="E6:49_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A16</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></span>;<br/><a NAME="E7:49_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A17</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E8:49_1_1_4_2"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_1_4_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E5:49_1_1_4_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E6:49_1_1_4_2"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">d1</font> = <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  as    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font>,1,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_1_4_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E7:49_1_1_4_2"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></span>;<br/><a NAME="E10:49_1_1_4_2"/><span class="lab"><font color="Green" title="E50">A18</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&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#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A19</font></span>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></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">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>
;<br/><a NAME="E12:49_1_1_4_2"/><span class="lab"><font color="Green" title="E52">A20</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1_1_4_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E3:49_1_1_4_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="matrix_2.html#T15" title="MATRIX_2:th.15">MATRIX_2:15</a></span>;<br/><span class="lab"><font color="Green" title="E53">A21</font></span>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_1_4_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E6:49_1_1_4_2"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1_1_4_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E2:49_1_1_4_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E3:49_1_1_4_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>
;<br/><a NAME="E14:49_1_1_4_2"/><span class="kw">then </span>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_1_4_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E5:49_1_1_4_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E6:49_1_1_4_2"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">d2</font> = <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  as    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of 1,1,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1_1_4_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></span>;<br/><span class="lab"><font color="Green" title="E54">A22</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Maroon" title="c8">d2</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49_1_1_4_2"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="matrlin.html#T32" target="_self" title="MATRLIN:th.32">Th32</a></span>
<br/>.= 
<span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_1_4_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E5:49_1_1_4_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E6:49_1_1_4_2"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E12:49_1_1_4_2"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="ref" href="matrix_2.html#T15" title="MATRIX_2:th.15">MATRIX_2:15</a></span>
<br/>.= 
<span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E5:49_1_1"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_1_1_4_2"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="txt" href="#E11:49_1_1_4_2"><span class="lab"><font color="Green" title="E51">A19</font></span></a>, <a class="ref" href="matrix_2.html#T15" title="MATRIX_2:th.15">MATRIX_2:15</a></span>
;<br/><span class="lab"><font color="Green" title="E55">A23</font></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"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1_1_4_2"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E3:49_1_1_4_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E11:49_1_1_4_2"><span class="lab"><font color="Green" title="E51">A19</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
;<br/><span class="kw">thus </span> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> = 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E5:49_1_1"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p4"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E5:49_1_1_4_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E2:49_1_1_4_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="matrlin.html#T35" target="_self" title="MATRLIN:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p5"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T19" target="_self" title="MATRLIN:th.19">Th19</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c7">d1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c8">d2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">d1</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Maroon" title="c8">d2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T31" target="_self" title="MATRLIN:th.31">Th31</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p5"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:49_1_1_4_2"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="txt" href="#E16:49_1_1_4_2"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="ref" href="matrix_2.html#T11" title="MATRIX_2:th.11">MATRIX_2:11</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E5:49_1"/>
<span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> is    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font>
 
;<br/>
<span class="kw">then </span> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T18" target="_self" title="MATRLIN:th.18">Th18</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font></span>)</span></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="matrlin.html#T17" target="_self" title="MATRLIN:th.17">Th17</a></span>
;<br/>
<a NAME="E7:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font>]
 
;<br/>
<a NAME="E8:49_1"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V1</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">P</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S2" title="FINSEQ_2:sch.2">FINSEQ_2:sch 2</a>(<a class="txt" href="#E7:49_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="txt" href="#E4:49_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:49_1"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">P</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></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><span class="kw">end;</span></div>
<a NAME="E2:49"/><span class="lab"><font color="Green" title="E37">A25</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="49_2"><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="Red">::  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="E1:49_2"/><span class="lab"><font color="Green" title="E38">A26</font></span>: 
for <font color="Olive" title="b1">M</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> holds <br/> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 ; <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="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">thus </span><a NAME="E2:49_2"/>
for <font color="Olive" title="b1">M</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M</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 holds <br/> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">M</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</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="c5">M</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  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:49_2_1"/><span class="lab"><font color="Green" title="E39">A27</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</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="Red">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span><br/>

<a NAME="E2:49_2_1"/><span class="lab"><font color="Green" title="E40">A28</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">M</font> <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:49_2_1_1"/>
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:49_2_1_1_1"/><span class="lab"><font color="Green" title="E41">A29</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:49_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2_1"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="txt" href="#E2:49_2_1"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">G</font> = <font color="Maroon" title="c5">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V1</font> ;<br/><a NAME="E4:49_2_1_1_1"/>
<font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c6">G</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2_1"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="txt" href="#E1:49_2_1_1_1"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><span class="kw">hence </span><a NAME="E5:49_2_1_1_1"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E36">A1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:49_2_1_1_2"/><span class="lab"><font color="Green" title="E41">A30</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:49_2_1_1_2"/><span class="lab"><font color="Green" title="E42">A31</font></span>: 
<font color="Maroon" title="c5">M</font> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M</font>,<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="#E1:49_2_1"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="txt" href="#E2:49_2_1"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">M1</font> = <font color="Maroon" title="c5">M</font> <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> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V1</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">R</font> = <span class="p1"><a href="matrlin.html#K2" title="MATRLIN:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c6">M1</font></span><a href="matrlin.html#K2" title="MATRLIN:func.2">*&gt;</a></span> as    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of 1, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_2_1_1_2"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">M9</font> = <font color="Maroon" title="c5">M</font> as    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2_1"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">W</font> =  <a href="matrlin.html#K1" title="MATRLIN:func.1">Del</a> <font color="Maroon" title="c8">M9</font>,<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> as    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c4">n</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T5" target="_self" title="MATRLIN:th.5">Th5</a></span>;<br/><span class="lab"><font color="Green" title="E43">A32</font></span>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">W</font> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c8">M9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2_1_1_2"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="ref" href="matrlin.html#T4" target="_self" title="MATRLIN:th.4">Th4</a></span>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">R</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T4" target="_self" title="MATRLIN:th.4">Th4</a></span>
;<br/><span class="lab"><font color="Green" title="E44">A33</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">W</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c9">W</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</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">R</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_2_1_1_2"><span class="lab"><font color="Green" title="E43">A32</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>
;<br/><a NAME="E9:49_2_1_1_2"/><span class="lab"><font color="Green" title="E45">A34</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K1" title="MATRLIN:func.1">Del</a> <font color="Maroon" title="c5">M</font>,<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> <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="#E1:49_2_1"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="matrlin.html#T3" target="_self" title="MATRLIN:th.3">Th3</a></span>;<br/><span class="kw">thus </span> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c5">M</font></span>)</span> = 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">W</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Maroon" title="c7">R</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2_1"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="matrlin.html#T6" target="_self" title="MATRLIN:th.6">Th6</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c9">W</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c7">R</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T31" target="_self" title="MATRLIN:th.31">Th31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p3">(<span class="default"><a href="matrlin.html#K1" title="MATRLIN:func.1">Del</a> <font color="Maroon" title="c5">M</font>,<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>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c7">R</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrlin.html#K1" title="MATRLIN:func.1">Del</a> <font color="Maroon" title="c5">M</font>,<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_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c7">R</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="txt" href="#E9:49_2_1_1_2"><span class="lab"><font color="Green" title="E45">A34</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrlin.html#K1" title="MATRLIN:func.1">Del</a> <font color="Maroon" title="c5">M</font>,<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_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E36">A1</font></span></a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">W</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrlin.html#K7" title="MATRLIN:func.7">^^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_2_1_1_2"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="ref" href="matrlin.html#T35" target="_self" title="MATRLIN:th.35">Th35</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">W</font> <a href="matrlin.html#K8" title="MATRLIN:func.8">^</a> <font color="Maroon" title="c7">R</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_2_1_1_2"><span class="lab"><font color="Green" title="E43">A32</font></span></a>, <a class="ref" href="matrlin.html#T32" target="_self" title="MATRLIN:th.32">Th32</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2_1"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="matrlin.html#T6" target="_self" title="MATRLIN:th.6">Th6</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E3:49"/><span class="lab"><font color="Green" title="E38">A35</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">M</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</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="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:49_3"/><span class="lab"><font color="Green" title="E39">A36</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span><br/>

<a NAME="E2:49_3"/><span class="kw">then </span>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></span>;<br/>
<a NAME="E3:49_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A37</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="c4">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<span class="kw">thus </span> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c4">M</font></span>)</span> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_3"><span class="lab"><font color="Green" title="E39">A36</font></span></a>, <a class="ref" href="matrlin.html#T17" target="_self" title="MATRLIN:th.17">Th17</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_3"><span class="lab"><font color="Green" title="E40">A37</font></span></a>, <a class="ref" href="matrlin.html#T17" target="_self" title="MATRLIN:th.17">Th17</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="E4:49"/>
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="#E3:49"><span class="lab"><font color="Green" title="E38">A35</font></span></a>, <a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E37">A25</font></span></a>);<br/></span>
<a NAME="E5:49"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font>]
 
;<br/>
<span class="kw">hence </span><a NAME="E6:49"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></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>
