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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c3">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c4">M</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">K</font>;<br/>







<b>assume </b><a NAME="E1:134"/><i><font color="Green" title="E109">A1</font></i>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">V</font> = <font color="Maroon" title="c1">n</font> <a href="matrix13.html#K9" title="MATRIX13:func.9">-VectSp_over</a> <font color="Maroon" title="c3">K</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">N</font> = <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1"/>
( <font color="Maroon" title="c2">m</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="c2">m</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="134_1_1"><b>suppose </b></a><a NAME="E1:134_1_1"/>
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:134_1_1"/><b>then </b>
 <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> 
 <b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><a NAME="E3:134_1_1"/><b>then </b>
 <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="matrix13.html#K9" title="MATRIX13:func.9">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/><b>hence </b><a NAME="E4:134_1_1"/>
 <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c4">M</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T4" title="VECTSP_7:th.4">VECTSP_7:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_2"><b>suppose </b></a><a NAME="E1:134_1_2"/><i><font color="Green" title="E110">A2</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:134_1_2"/><b>then </b><i><font color="Green" title="E111">A3</font></i>: 
 <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> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T1" target="_self" title="MATRIX13:th.1">Th1</a></i>;<br/><a NAME="E3:134_1_2"/><i><font color="Green" title="E112">A4</font></i>: 
<font color="Maroon" title="c4">M</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">without_repeated_line</a>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:134"><i><font color="Green" title="E109">A1</font></i></a>, <a class="ref" href="matrix13.html#T105" target="_self" title="MATRIX13:th.105">Th105</a></i>;<br/><div><i><font color="Green" title="E113">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:134_1_2_1"/><i><font color="Green" title="E114">A6</font></i>: 
<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> <font color="Maroon" title="c2">m</font>
 ;<br/><b>assume </b><a NAME="E2:134_1_2_1"/><i><font color="Green" title="E115">A7</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span>
 ;<br/><a NAME="E3:134_1_2_1"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">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="c4">M</font></span>)</span> &amp;  <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> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:134_1_2_1"/><i><font color="Green" title="E116">A8</font></i>: 
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:134_1_2_1"><i><font color="Green" title="E114">A6</font></i></a>, <a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></i>;<br/><a NAME="E6:134_1_2_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:134"><i><font color="Green" title="E109">A1</font></i></a>, <a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E2:134_1_2_1"><i><font color="Green" title="E115">A7</font></i></a>, <a class="ref" href="matrix13.html#T90" target="_self" title="MATRIX13:th.90">Th90</a></i>;<br/><a NAME="E7:134_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E5:134_1_2_1"><i><font color="Green" title="E116">A8</font></i></a>, <a class="ref" href="matrix13.html#T74" target="_self" title="MATRIX13:th.74">Th74</a></i>;<br/><b>hence </b><a NAME="E8:134_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix13.html#E5:134_1_2_1"><i><font color="Green" title="E116">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">M1</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">K</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:134_1_2_2"/><i><font color="Green" title="E114">A9</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> holds <br/> ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font> st  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Olive" title="b1">i</font></span>)</span>
 <b>and </b><br/><a NAME="E2:134_1_2_2"/><i><font color="Green" title="E115">A10</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c7">M1</font>,<font color="Olive" title="b1">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font>
 ;<br/><b>set </b><font color="Maroon" title="c8">NULL</font> =  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>;<br/><a NAME="E3:134_1_2_2"/><i><font color="Green" title="E116">A11</font></i>: 
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">M</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:134_1_2"><i><font color="Green" title="E110">A2</font></i></a>, <a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="ref" href="matrix13.html#T1" target="_self" title="MATRIX13:th.1">Th1</a>, <a class="ref" href="matrix_1.html#T26" title="MATRIX_1:th.26">MATRIX_1:26</a>, <a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></i>;<br/><b>assume </b><a NAME="E4:134_1_2_2"/>
<font color="Maroon" title="c7">M1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:134_1_2_2"/><i><font color="Green" title="E117">A12</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c7">M1</font> &amp; <font color="Maroon" title="c7">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">i</font> = <font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c12">j</font> = <font color="Maroon" title="c10">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E8:134_1_2_2"/><i><font color="Green" title="E118">A13</font></i>: 
( <font color="Maroon" title="c7">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:134_1_2_2"><i><font color="Green" title="E116">A11</font></i></a>, <a class="txt" href="matrix13.html#E6:134_1_2_2"><i><font color="Green" title="E117">A12</font></i></a>, <a class="ref" href="matrix_3.html#T3" title="MATRIX_3:th.3">MATRIX_3:3</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font><b> such that </b><br/><a NAME="E10:134_1_2_2"/><i><font color="Green" title="E119">A14</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:134_1_2_2"><i><font color="Green" title="E114">A9</font></i></a></i>;<br/><a NAME="E11:134_1_2_2"/><i><font color="Green" title="E120">A15</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><b>set </b><font color="Maroon" title="c14">R</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span>;<br/><font color="Maroon" title="c7">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
<b>by </b><i><a class="txt" href="matrix13.html#E3:134_1_2_2"><i><font color="Green" title="E116">A11</font></i></a>, <a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c13">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="txt" href="matrix13.html#E10:134_1_2_2"><i><font color="Green" title="E119">A14</font></i></a>, <a class="txt" href="matrix13.html#E11:134_1_2_2"><i><font color="Green" title="E120">A15</font></i></a>, <a class="ref" href="fvsum_1.html#T63" title="FVSUM_1:th.63">FVSUM_1:63</a></i>
;<br/><a NAME="E13:134_1_2_2"/><b>then </b>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/><a NAME="E14:134_1_2_2"/><b>then </b><i><font color="Green" title="E121">A16</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:134"><i><font color="Green" title="E109">A1</font></i></a>, <a class="ref" href="matrix13.html#T89" target="_self" title="MATRIX13:th.89">Th89</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">L</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c4">M</font><b> such that </b><br/><a NAME="E16:134_1_2_2"/><i><font color="Green" title="E122">A17</font></i>: 
<font color="Maroon" title="c15">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p1">(<span class="default"><a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M1</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:134_1_2"><i><font color="Green" title="E112">A4</font></i></a>, <a class="txt" href="matrix13.html#E1:134_1_2_2"><i><font color="Green" title="E114">A9</font></i></a>, <a class="ref" href="matrix13.html#T108" target="_self" title="MATRIX13:th.108">Th108</a></i>;<br/><b>set </b><font color="Maroon" title="c16">LM</font> = <font color="Maroon" title="c15">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p1">(<span class="default"><a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <font color="Maroon" title="c4">M</font></span>)</span>;<br/><a NAME="E17:134_1_2_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M1</font> <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="c4">M</font> &amp;  <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> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E16:134_1_2_2"><i><font color="Green" title="E122">A17</font></i></a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a>, <a class="ref" href="vectsp_6.html#D8" title="VECTSP_6:def.8">VECTSP_6:def 8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="matrix13.html#K9" title="MATRIX13:func.9">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span><b> such that </b><br/><a NAME="E19:134_1_2_2"/><i><font color="Green" title="E123">A18</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">m</font>
 <b>and </b><br/><a NAME="E20:134_1_2_2"/><i><font color="Green" title="E124">A19</font></i>: 
<font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="matrix13.html#K9" title="MATRIX13:func.9">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <b>and </b><br/><a NAME="E21:134_1_2_2"/><i><font color="Green" title="E125">A20</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E16:134_1_2_2"><i><font color="Green" title="E122">A17</font></i></a>, <a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font> implies for <font color="Olive" title="b1">t</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span> );<br/> <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
;<br/><a NAME="E23:134_1_2_2"/><b>then </b><i><font color="Green" title="E126">A21</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M</font> )
 <b>by </b><i><a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E24:134_1_2_2"/><i><font color="Green" title="E127">A22</font></i>: 
<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="134_1_2_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:134_1_2_2_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c18">t</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font>;<br/>

<b>assume </b><a NAME="E2:134_1_2_2_3"/><i><font color="Green" title="E128">A23</font></i>: 
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E3:134_1_2_2_3"/>
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix13.html#E20:134_1_2_2"><i><font color="Green" title="E124">A19</font></i></a>, <a class="txt" href="matrix13.html#E2:134_1_2_2_3"><i><font color="Green" title="E128">A23</font></i></a>, <a class="ref" href="matrix13.html#T102" target="_self" title="MATRIX13:th.102">Th102</a></i>;<br/>
<a NAME="E4:134_1_2_2_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font> &amp;  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E23:134_1_2_2"><i><font color="Green" title="E126">A21</font></i></a>, <a class="ref" href="fvsum_1.html#T28" title="FVSUM_1:th.28">FVSUM_1:28</a>, <a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a></i>;<br/>
<b>hence </b><a NAME="E5:134_1_2_2_3"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E25:134_1_2_2"/><i><font color="Green" title="E128">A24</font></i>: 
for <font color="Olive" title="b1">k</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">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_2_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:134_1_2_2_4"/><i><font color="Green" title="E129">A25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">k</font>]
 ;<br/>

<b>set </b><font color="Maroon" title="c19">k1</font> = <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>assume </b><a NAME="E2:134_1_2_2_4"/><i><font color="Green" title="E130">A26</font></i>: 
<font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c20">t</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font>;<br/>

<b>assume </b><a NAME="E3:134_1_2_2_4"/><i><font color="Green" title="E131">A27</font></i>: 
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c21">kk</font> = <font color="Maroon" title="c18">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">t1</font> = <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">kk</font>, <font color="Maroon" title="c23">T</font> = <font color="Maroon" title="c20">t</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c6">N</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> <b>by </b><i><a class="ref" href="matrix13.html#T102" target="_self" title="MATRIX13:th.102">Th102</a></i>;<br/>
<b>set </b><font color="Maroon" title="c24">RR</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c22">t1</font></span>)</span>;<br/>
<a NAME="E6:134_1_2_2_4"/>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="txt" href="matrix13.html#E2:134_1_2_2_4"><i><font color="Green" title="E130">A26</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:134_1_2_2_4"/><b>then </b><i><font color="Green" title="E132">A28</font></i>: 
(  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c22">t1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E1:134_1_2_2_4"><i><font color="Green" title="E129">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:134_1_2_2_4"/>
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:134_1_2_2_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2_2_4"><i><font color="Green" title="E130">A26</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:134_1_2_2_4"/><b>then </b><i><font color="Green" title="E133">A29</font></i>: 
<font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c25">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font><b> such that </b><br/><a NAME="E12:134_1_2_2_4"/><i><font color="Green" title="E134">A30</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</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> <font color="Maroon" title="c25">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:134_1_2_2"><i><font color="Green" title="E114">A9</font></i></a></i>;<br/>
<a NAME="E13:134_1_2_2_4"/><i><font color="Green" title="E135">A31</font></i>: 
(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</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> <font color="Maroon" title="c7">M1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c7">M1</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E10:134_1_2_2_4"><i><font color="Green" title="E133">A29</font></i></a>, <a class="ref" href="matrix13.html#T103" target="_self" title="MATRIX13:th.103">Th103</a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">LMk1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c15">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="matrix13.html#K9" title="MATRIX13:func.9">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span> <b>by </b><i><a class="txt" href="matrix13.html#E16:134_1_2_2"><i><font color="Green" title="E122">A17</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">LR</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c28">LM1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c6">N</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> <b>by </b><i><a class="txt" href="matrix13.html#E1:134_1_2"><i><font color="Green" title="E110">A2</font></i></a>, <a class="ref" href="matrix13.html#T1" target="_self" title="MATRIX13:th.1">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">LRt</font> = <font color="Maroon" title="c27">LR</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c20">t</font>, <font color="Maroon" title="c30">LRt1</font> = <font color="Maroon" title="c27">LR</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c22">t1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E17:134_1_2_2_4"/><i><font color="Green" title="E136">A32</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">LR</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c22">t1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E23:134_1_2_2"><i><font color="Green" title="E126">A21</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E18:134_1_2_2_4"/><b>then </b><i><font color="Green" title="E137">A33</font></i>: 
(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c22">t1</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">LR</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c22">t1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c22">t1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c22">t1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a>, <a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/>
<i><font color="Green" title="E138">A34</font></i>:  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2_2_4"><i><font color="Green" title="E130">A26</font></i></a>, <a class="txt" href="matrix13.html#E10:134_1_2_2_4"><i><font color="Green" title="E133">A29</font></i></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></i>
<br/>.= 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c22">t1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2_2_4"><i><font color="Green" title="E130">A26</font></i></a>, <a class="txt" href="matrix13.html#E10:134_1_2_2_4"><i><font color="Green" title="E133">A29</font></i></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></i>
;<br/>
<a NAME="E20:134_1_2_2_4"/><i><font color="Green" title="E139">A35</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">LR</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c23">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<b>then </b><i><font color="Green" title="E140">A36</font></i>:  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c22">t1</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c27">LR</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c20">t</font></span>)</span> = 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c22">t1</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c29">LRt</font>
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E23:134_1_2_2"><i><font color="Green" title="E126">A21</font></i></a>, <a class="txt" href="matrix13.html#E18:134_1_2_2_4"><i><font color="Green" title="E137">A33</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c30">LRt1</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c29">LRt</font>
<b>by </b><i><a class="txt" href="matrix13.html#E17:134_1_2_2_4"><i><font color="Green" title="E136">A32</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c29">LRt</font>
<b>by </b><i><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></i>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c27">LR</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c20">t</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E23:134_1_2_2"><i><font color="Green" title="E126">A21</font></i></a>, <a class="txt" href="matrix13.html#E20:134_1_2_2_4"><i><font color="Green" title="E139">A35</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
;<br/>
<font color="Maroon" title="c20">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">kk</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c26">LMk1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E21:134_1_2_2"><i><font color="Green" title="E125">A20</font></i></a>, <a class="txt" href="matrix13.html#E3:134_1_2_2_4"><i><font color="Green" title="E131">A27</font></i></a>, <a class="txt" href="matrix13.html#E7:134_1_2_2_4"><i><font color="Green" title="E132">A28</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c22">t1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E3:134_1_2_2"><i><font color="Green" title="E116">A11</font></i></a>, <a class="txt" href="matrix13.html#E16:134_1_2_2"><i><font color="Green" title="E122">A17</font></i></a>, <a class="txt" href="matrix13.html#E13:134_1_2_2_4"><i><font color="Green" title="E135">A31</font></i></a>, <a class="ref" href="matrix13.html#T102" target="_self" title="MATRIX13:th.102">Th102</a></i>
;<br/>
<b>then </b><font color="Maroon" title="c27">LR</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c20">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">LR</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c22">t1</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c28">LM1</font>
<b>by </b><i><a class="ref" href="finseqop.html#T29" title="FINSEQOP:th.29">FINSEQOP:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c22">t1</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p0">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c22">t1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="txt" href="matrix13.html#E12:134_1_2_2_4"><i><font color="Green" title="E134">A30</font></i></a>, <a class="txt" href="matrix13.html#E17:134_1_2_2_4"><i><font color="Green" title="E136">A32</font></i></a>, <a class="txt" href="matrix13.html#E19:134_1_2_2_4"><i><font color="Green" title="E138">A34</font></i></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></i>
;<br/>
<b>hence </b><a NAME="E24:134_1_2_2_4"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c20">t</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2_2_4"><i><font color="Green" title="E130">A26</font></i></a>, <a class="txt" href="matrix13.html#E7:134_1_2_2_4"><i><font color="Green" title="E132">A28</font></i></a>, <a class="txt" href="matrix13.html#E10:134_1_2_2_4"><i><font color="Green" title="E133">A29</font></i></a>, <a class="txt" href="matrix13.html#E18:134_1_2_2_4"><i><font color="Green" title="E137">A33</font></i></a>, <a class="txt" href="matrix13.html#E21:134_1_2_2_4"><i><font color="Green" title="E140">A36</font></i></a>, <a class="ref" href="matrix13.html#T92" target="_self" title="MATRIX13:th.92">Th92</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:134_1_2_2"/><i><font color="Green" title="E129">A37</font></i>: 
for <font color="Olive" title="b1">k</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">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="matrix13.html#E24:134_1_2_2"><i><font color="Green" title="E127">A22</font></i></a>, <a class="txt" href="matrix13.html#E25:134_1_2_2"><i><font color="Green" title="E128">A24</font></i></a>);<br/></i><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> implies for <font color="Olive" title="b1">t</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 holds <br/> <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">t</font></span>)</span> );<br/><a NAME="E27:134_1_2_2"/><i><font color="Green" title="E130">A38</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E28:134_1_2_2"/><i><font color="Green" title="E131">A39</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_2_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:134_1_2_2_5"/><i><font color="Green" title="E132">A40</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">k</font>]
 ;<br/>

<b>set </b><font color="Maroon" title="c19">k1</font> = <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>assume </b><a NAME="E2:134_1_2_2_5"/><i><font color="Green" title="E133">A41</font></i>: 
( <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c20">t</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font>;<br/>

<b>assume </b><a NAME="E3:134_1_2_2_5"/><i><font color="Green" title="E134">A42</font></i>: 
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c21">kk</font> = <font color="Maroon" title="c18">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">t1</font> = <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">kk</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c6">N</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> <b>by </b><i><a class="ref" href="matrix13.html#T102" target="_self" title="MATRIX13:th.102">Th102</a></i>;<br/>
<a NAME="E6:134_1_2_2_5"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E7:134_1_2_2_5"/><b>then </b><i><font color="Green" title="E135">A43</font></i>: 
<font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2_2_5"><i><font color="Green" title="E133">A41</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font><b> such that </b><br/><a NAME="E9:134_1_2_2_5"/><i><font color="Green" title="E136">A44</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</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> <font color="Maroon" title="c23">b</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:134_1_2_2"><i><font color="Green" title="E114">A9</font></i></a></i>;<br/>
<a NAME="E10:134_1_2_2_5"/><i><font color="Green" title="E137">A45</font></i>: 
(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</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> <font color="Maroon" title="c7">M1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c7">M1</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E7:134_1_2_2_5"><i><font color="Green" title="E135">A43</font></i></a>, <a class="ref" href="matrix13.html#T103" target="_self" title="MATRIX13:th.103">Th103</a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c24">LMk1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c15">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="matrix13.html#K9" title="MATRIX13:func.9">-VectSp_over</a> <font color="Maroon" title="c3">K</font></span>)</span> <b>by </b><i><a class="txt" href="matrix13.html#E16:134_1_2_2"><i><font color="Green" title="E122">A17</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">LR</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c26">LM1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c7">M1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> <b>by </b><i><a class="txt" href="matrix13.html#E1:134_1_2"><i><font color="Green" title="E110">A2</font></i></a>, <a class="ref" href="matrix13.html#T1" target="_self" title="MATRIX13:th.1">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">T</font> = <font color="Maroon" title="c20">t</font>, <font color="Maroon" title="c28">T1</font> = <font color="Maroon" title="c22">t1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E14:134_1_2_2_5"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2_2_5"><i><font color="Green" title="E133">A41</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>then </b><i><font color="Green" title="E138">A46</font></i>: <font color="Maroon" title="c20">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">kk</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c24">LMk1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E21:134_1_2_2"><i><font color="Green" title="E125">A20</font></i></a>, <a class="txt" href="matrix13.html#E3:134_1_2_2_5"><i><font color="Green" title="E134">A42</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c22">t1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c26">LM1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E16:134_1_2_2"><i><font color="Green" title="E122">A17</font></i></a>, <a class="txt" href="matrix13.html#E10:134_1_2_2_5"><i><font color="Green" title="E137">A45</font></i></a>, <a class="ref" href="matrix13.html#T102" target="_self" title="MATRIX13:th.102">Th102</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_2_2_5_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1_2_2_5_2"/>
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2_2_5"><i><font color="Green" title="E133">A41</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_2_2_5_2_1"><b>suppose </b></a><a NAME="E1:134_1_2_2_5_2_1"/><i><font color="Green" title="E139">A47</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:134_1_2_2_5_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">LM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E3:134_1_2_2_5_2_1"/><b>then </b>
<font color="Maroon" title="c25">LR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">LM1</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E10:134_1_2_2"><i><font color="Green" title="E119">A14</font></i></a>, <a class="txt" href="matrix13.html#E7:134_1_2_2_5"><i><font color="Green" title="E135">A43</font></i></a>, <a class="txt" href="matrix13.html#E1:134_1_2_2_5_2_1"><i><font color="Green" title="E139">A47</font></i></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></i>;<br/><a NAME="E4:134_1_2_2_5_2_1"/><b>then </b><i><font color="Green" title="E140">A48</font></i>: 
<font color="Maroon" title="c25">LR</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c22">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">t</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E15:134_1_2_2_5"><i><font color="Green" title="E138">A46</font></i></a>, <a class="ref" href="finseqop.html#T34" title="FINSEQOP:th.34">FINSEQOP:34</a></i>;<br/><a NAME="E5:134_1_2_2_5_2_1"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:134_1_2_2_5_2_1"><i><font color="Green" title="E139">A47</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E6:134_1_2_2_5_2_1"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c20">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E26:134_1_2_2"><i><font color="Green" title="E129">A37</font></i></a>, <a class="txt" href="matrix13.html#E4:134_1_2_2_5_2_1"><i><font color="Green" title="E140">A48</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_2_2_5_2_2"><b>suppose </b></a><a NAME="E1:134_1_2_2_5_2_2"/><i><font color="Green" title="E139">A49</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:134_1_2_2_5_2_2"/><b>then </b><i><font color="Green" title="E140">A50</font></i>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c22">t1</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:134_1_2_2_5"><i><font color="Green" title="E132">A40</font></i></a>, <a class="txt" href="matrix13.html#E2:134_1_2_2_5"><i><font color="Green" title="E133">A41</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c29">RR</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c22">t1</font>;<br/><a NAME="E3:134_1_2_2_5_2_2"/><i><font color="Green" title="E141">A51</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E23:134_1_2_2"><i><font color="Green" title="E126">A21</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E4:134_1_2_2_5_2_2"/><b>then </b><i><font color="Green" title="E142">A52</font></i>: 
(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c22">t1</font></span>)</span>,<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">t1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c22">t1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c22">t1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a>, <a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E143">A53</font></i>: <font color="Maroon" title="c20">t</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c22">t1</font></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">b</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E7:134_1_2_2_5"><i><font color="Green" title="E135">A43</font></i></a>, <a class="txt" href="matrix13.html#E9:134_1_2_2_5"><i><font color="Green" title="E136">A44</font></i></a>, <a class="txt" href="matrix13.html#E15:134_1_2_2_5"><i><font color="Green" title="E138">A46</font></i></a>, <a class="txt" href="matrix13.html#E1:134_1_2_2_5_2_2"><i><font color="Green" title="E139">A49</font></i></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c22">t1</font></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">b</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p4">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p0">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c22">t1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E7:134_1_2_2_5"><i><font color="Green" title="E135">A43</font></i></a>, <a class="txt" href="matrix13.html#E1:134_1_2_2_5_2_2"><i><font color="Green" title="E139">A49</font></i></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a></i>
;<br/><a NAME="E6:134_1_2_2_5_2_2"/><i><font color="Green" title="E144">A54</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><b>then </b><i><font color="Green" title="E145">A55</font></i>:  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c22">t1</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c20">t</font> = 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c22">t1</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c27">T</font>
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E23:134_1_2_2"><i><font color="Green" title="E126">A21</font></i></a>, <a class="txt" href="matrix13.html#E4:134_1_2_2_5_2_2"><i><font color="Green" title="E142">A52</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c28">T1</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c27">T</font>
<b>by </b><i><a class="txt" href="matrix13.html#E3:134_1_2_2_5_2_2"><i><font color="Green" title="E141">A51</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c27">T</font>
<b>by </b><i><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></i>
<br/>.= 
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c20">t</font>
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E23:134_1_2_2"><i><font color="Green" title="E126">A21</font></i></a>, <a class="txt" href="matrix13.html#E6:134_1_2_2_5_2_2"><i><font color="Green" title="E144">A54</font></i></a>, <a class="ref" href="matrix11.html#T29" title="MATRIX11:th.29">MATRIX11:29</a></i>
;<br/><a NAME="E8:134_1_2_2_5_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c22">t1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><b>hence </b><a NAME="E9:134_1_2_2_5_2_2"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c20">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E7:134_1_2_2_5"><i><font color="Green" title="E135">A43</font></i></a>, <a class="txt" href="matrix13.html#E1:134_1_2_2_5_2_2"><i><font color="Green" title="E139">A49</font></i></a>, <a class="txt" href="matrix13.html#E2:134_1_2_2_5_2_2"><i><font color="Green" title="E140">A50</font></i></a>, <a class="txt" href="matrix13.html#E5:134_1_2_2_5_2_2"><i><font color="Green" title="E143">A53</font></i></a>, <a class="txt" href="matrix13.html#E7:134_1_2_2_5_2_2"><i><font color="Green" title="E145">A55</font></i></a>, <a class="ref" href="matrix13.html#T92" target="_self" title="MATRIX13:th.92">Th92</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E29:134_1_2_2"/><i><font color="Green" title="E132">A56</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="matrix13.html#E27:134_1_2_2"><i><font color="Green" title="E130">A38</font></i></a>, <a class="txt" href="matrix13.html#E28:134_1_2_2"><i><font color="Green" title="E131">A39</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c18">SumLM</font> =  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> <b>by </b><i><a class="ref" href="matrix13.html#T102" target="_self" title="MATRIX13:th.102">Th102</a></i>;<br/><b>set </b><font color="Maroon" title="c19">n0</font> = <font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span>;<br/><a NAME="E31:134_1_2_2"/><i><font color="Green" title="E133">A57</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">SumLM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_2_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:134_1_2_2_6"/><i><font color="Green" title="E134">A58</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 ;<br/><a NAME="E2:134_1_2_2_6"/>
<font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:134_1_2_2_6"/><b>then </b><i><font color="Green" title="E135">A59</font></i>: 
( <font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c15">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c4">M</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:134_1_2_2_6"><i><font color="Green" title="E134">A58</font></i></a>, <a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/><a NAME="E4:134_1_2_2_6"/>
<font color="Maroon" title="c7">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K11" title="MATRIX13:func.11">FinS2MX</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E16:134_1_2_2"><i><font color="Green" title="E122">A17</font></i></a></i>;<br/><b>then </b> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c7">M1</font>,<font color="Maroon" title="c20">j</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c15">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font>
<b>by </b><i><a class="txt" href="matrix13.html#E1:134"><i><font color="Green" title="E109">A1</font></i></a>, <a class="txt" href="matrix13.html#E3:134_1_2_2_6"><i><font color="Green" title="E135">A59</font></i></a>, <a class="ref" href="matrix13.html#T105" target="_self" title="MATRIX13:th.105">Th105</a>, <a class="ref" href="matrix13.html#T107" target="_self" title="MATRIX13:th.107">Th107</a></i>
<br/>.= 
<font color="Maroon" title="c18">SumLM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font>
<b>by </b><i><a class="txt" href="matrix13.html#E3:134_1_2"><i><font color="Green" title="E112">A4</font></i></a>, <a class="txt" href="matrix13.html#E3:134_1_2_2_6"><i><font color="Green" title="E135">A59</font></i></a>, <a class="ref" href="vectsp_9.html#T7" title="VECTSP_9:th.7">VECTSP_9:7</a></i>
;<br/><b>hence </b><font color="Maroon" title="c18">SumLM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font>
<b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2_2"><i><font color="Green" title="E115">A10</font></i></a>, <a class="txt" href="matrix13.html#E3:134_1_2_2_6"><i><font color="Green" title="E135">A59</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font>
<b>by </b><i><a class="txt" href="matrix13.html#E3:134_1_2_2_6"><i><font color="Green" title="E135">A59</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_2.html#T71" title="FINSEQ_2:th.71">FINSEQ_2:71</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E33:134_1_2_2"/><b>then </b>
( <font color="Maroon" title="c18">SumLM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="txt" href="matrix13.html#E31:134_1_2_2"><i><font color="Green" title="E133">A57</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/><a NAME="E34:134_1_2_2"/><b>then </b><i><font color="Green" title="E134">A60</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E14:134_1_2_2"><i><font color="Green" title="E121">A16</font></i></a>, <a class="txt" href="matrix13.html#E19:134_1_2_2"><i><font color="Green" title="E123">A18</font></i></a>, <a class="txt" href="matrix13.html#E29:134_1_2_2"><i><font color="Green" title="E132">A56</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c20">RR</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>;<br/><a NAME="E35:134_1_2_2"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</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">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E37:134_1_2_2"/><i><font color="Green" title="E135">A61</font></i>: 
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">k</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></i>;<br/><a NAME="E38:134_1_2_2"/>
(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E8:134_1_2_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="txt" href="matrix13.html#E23:134_1_2_2"><i><font color="Green" title="E126">A21</font></i></a>, <a class="txt" href="matrix13.html#E31:134_1_2_2"><i><font color="Green" title="E133">A57</font></i></a>, <a class="ref" href="matrix11.html#T28" title="MATRIX11:th.28">MATRIX11:28</a>, <a class="ref" href="matrix11.html#D3" title="MATRIX11:def.3">MATRIX11:def 3</a></i>;<br/><a NAME="E39:134_1_2_2"/><b>then </b>
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E2:134_1_2"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="matrix13.html#E23:134_1_2_2"><i><font color="Green" title="E126">A21</font></i></a>, <a class="txt" href="matrix13.html#E34:134_1_2_2"><i><font color="Green" title="E134">A60</font></i></a>, <a class="ref" href="matrix13.html#T90" target="_self" title="MATRIX13:th.90">Th90</a></i>;<br/><a NAME="E40:134_1_2_2"/><b>then </b>
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">k</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E37:134_1_2_2"><i><font color="Green" title="E135">A61</font></i></a>, <a class="ref" href="matrix13.html#T74" target="_self" title="MATRIX13:th.74">Th74</a></i>;<br/><b>hence </b><a NAME="E41:134_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix13.html#E37:134_1_2_2"><i><font color="Green" title="E135">A61</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:134_1_2"/>
 <a href="matrix13.html#K10" title="MATRIX13:func.10">lines</a> <font color="Maroon" title="c4">M</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:134_1_2"><i><font color="Green" title="E112">A4</font></i></a>, <a class="txt" href="matrix13.html#E4:134_1_2"><i><font color="Green" title="E113">A5</font></i></a>, <a class="ref" href="matrix13.html#T109" target="_self" title="MATRIX13:th.109">Th109</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
