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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c5">L</font> be    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c4">K</font>;<br/><b>let </b><font color="Maroon" title="c6">M</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">K</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:130"/><i><font color="Green" title="E106">A1</font></i>: 
( <font color="Maroon" title="c6">M</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">without_repeated_line</a> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">lines</a> <font color="Maroon" title="c6">M</font> )
 <b>and </b><br/><a NAME="E2:130"/><i><font color="Green" title="E107">A2</font></i>: 
<font color="Maroon" title="c2">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="c3">n</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">V</font> = <font color="Maroon" title="c3">n</font> <a href="matrix13.html#K7" title="MATRIX13:func.7">-VectSp_over</a> <font color="Maroon" title="c4">K</font>;<br/>
<b>set </b><font color="Maroon" title="c8">MX</font> =  <a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font>;<br/>
<b>set </b><font color="Maroon" title="c9">LM</font> = <font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">FLM</font> =  <a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">C</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p3">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">i</font>;<br/>
<a NAME="E3:130"/><i><font color="Green" title="E108">A3</font></i>: 
(  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E1:130"><i><font color="Green" title="E106">A1</font></i></a>, <a class="ref" href="vectsp_6.html#D8" title="VECTSP_6:def.8">VECTSP_6:def 8</a>, <a class="ref" href="vectsp_9.html#T7" title="VECTSP_9:th.7">VECTSP_9:7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">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="c3">n</font> <a href="matrix13.html#K7" title="MATRIX13:func.7">-VectSp_over</a> <font color="Maroon" title="c4">K</font></span>)</span><b> such that </b><br/><a NAME="E5:130"/><i><font color="Green" title="E109">B1</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font></span>)</span>
 <b>and </b><br/><a NAME="E6:130"/><i><font color="Green" title="E110">B2</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <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="c3">n</font> <a href="matrix13.html#K7" title="MATRIX13:func.7">-VectSp_over</a> <font color="Maroon" title="c4">K</font></span>)</span>
 <b>and </b><br/><a NAME="E7:130"/><i><font color="Green" title="E111">B3</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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="c3">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c4">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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">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="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c12">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D12" title="RLVECT_1:def.12">RLVECT_1:def 12</a></i>;<br/>
<a NAME="E8:130"/><i><font color="Green" title="E112">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p4">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">g</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 <font color="Maroon" title="c4">K</font><b> such that </b><br/><a NAME="E10:130"/><i><font color="Green" title="E113">D1</font></i>: 
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p4">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font></span>)</span>
 <b>and </b><br/><a NAME="E11:130"/><i><font color="Green" title="E114">D2</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <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="c4">K</font>
 <b>and </b><br/><a NAME="E12:130"/><i><font color="Green" title="E115">D3</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <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="c4">K</font>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p4">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">i</font></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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c13">g</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">a</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:130"><i><font color="Green" title="E108">A3</font></i></a>, <a class="ref" href="rlvect_1.html#D12" title="RLVECT_1:def.12">RLVECT_1:def 12</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#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font> implies for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="matrix13.html#K7" title="MATRIX13:func.7">-VectSp_over</a> <font color="Maroon" title="c4">K</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 holds <br/><font color="Olive" title="b1">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E13:130"/><i><font color="Green" title="E116">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:130_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c14">v</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c4">K</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:130_1"/><i><font color="Green" title="E117">A6</font></i>: 
<font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0
 ;<br/>

<a NAME="E3:130_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="matrix13.html#K7" title="MATRIX13:func.7">-VectSp_over</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c4">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix13.html#T102" target="_self" title="MATRIX13:th.102">TT102</a></i>;<br/>
<b>hence </b><a NAME="E4:130_1"/>
<font color="Maroon" title="c14">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0
 <b>by </b><i><a class="txt" href="matrix13.html#E11:130"><i><font color="Green" title="E114">D2</font></i></a>, <a class="txt" href="matrix13.html#E2:130"><i><font color="Green" title="E107">A2</font></i></a>, <a class="txt" href="matrix13.html#E6:130"><i><font color="Green" title="E110">B2</font></i></a>, <a class="txt" href="matrix13.html#E2:130_1"><i><font color="Green" title="E117">A6</font></i></a>, <a class="ref" href="finseq_2.html#T71" title="FINSEQ_2:th.71">FINSEQ_2:71</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:130"/><i><font color="Green" title="E117">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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="130_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:130_2"/><i><font color="Green" title="E118">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font>]
 ;<br/>

<b>set </b><font color="Maroon" title="c15">k1</font> = <font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon" title="c16">K1</font> = <font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>assume </b><a NAME="E3:130_2"/><i><font color="Green" title="E119">A8</font></i>: 
<font color="Maroon" title="c14">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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c17">v</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c4">K</font></span>)</span>;<br/>

<b>assume </b><a NAME="E4:130_2"/><i><font color="Green" title="E120">A9</font></i>: 
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E5:130_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">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="E6:130_2"/><b>then </b><i><font color="Green" title="E121">A10</font></i>: 
<font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix13.html#E3:130_2"><i><font color="Green" title="E119">A8</font></i></a></i>;<br/>
<a NAME="E7:130_2"/><b>then </b><i><font color="Green" title="E122">A11</font></i>: 
( <font color="Maroon" title="c14">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span> &amp; <font color="Maroon" title="c14">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p3">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E3:130"><i><font color="Green" title="E108">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:130_2"/><i><font color="Green" title="E123">A12</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c16">K1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p3">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E3:130"><i><font color="Green" title="E108">A3</font></i></a>, <a class="txt" href="matrix13.html#E8:130"><i><font color="Green" title="E112">A4</font></i></a>, <a class="txt" href="matrix13.html#E6:130_2"><i><font color="Green" title="E121">A10</font></i></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a>, <a class="ref" href="matrix13.html#T1" target="_self" title="MATRIX13:th.1">TT16</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E9:130_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">lines</a> <font color="Maroon" title="c6">M</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E8:130"><i><font color="Green" title="E112">A4</font></i></a>, <a class="txt" href="matrix13.html#E6:130_2"><i><font color="Green" title="E121">A10</font></i></a>, <a class="ref" href="matrix13.html#T103" target="_self" title="MATRIX13:th.103">TT103</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">MXK1</font> = <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">K1</font> 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="c3">n</font> <a href="matrix13.html#K7" title="MATRIX13:func.7">-VectSp_over</a> <font color="Maroon" title="c4">K</font></span>)</span> ;<br/>
<a NAME="E11:130_2"/>
<span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">K1</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E7:130_2"><i><font color="Green" title="E122">A11</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E12:130_2"/><b>then </b><i><font color="Green" title="E124">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">MXK1</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c18">MXK1</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E7:130_2"><i><font color="Green" title="E122">A11</font></i></a>, <a class="ref" href="vectsp_6.html#D8" title="VECTSP_6:def.8">VECTSP_6:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">LMK1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p2">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">K1</font> 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="c3">n</font> <a href="matrix13.html#K7" title="MATRIX13:func.7">-VectSp_over</a> <font color="Maroon" title="c4">K</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c20">lmk1</font> = <font color="Maroon" title="c19">LMK1</font>, <font color="Maroon" title="c21">mxk1</font> = <font color="Maroon" title="c18">MXK1</font>, <font color="Maroon" title="c22">fk</font> = <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">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="c4">K</font> <b>by </b><i><a class="ref" href="matrix13.html#T102" target="_self" title="MATRIX13:th.102">TT102</a></i>;<br/>
<a NAME="E15:130_2"/>
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c20">lmk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></i>;<br/>
<a NAME="E16:130_2"/><b>then </b>
( <font color="Maroon" title="c20">lmk1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">lmk1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">lmk1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">K</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:130"><i><font color="Green" title="E107">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c23">lmk1i</font> = <font color="Maroon" title="c20">lmk1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">K</font> ;<br/>
<font color="Maroon" title="c19">LMK1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">MXK1</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <font color="Maroon" title="c21">mxk1</font>
<b>by </b><i><a class="ref" href="matrix13.html#T102" target="_self" title="MATRIX13:th.102">TT102</a>, <a class="txt" href="matrix13.html#E12:130_2"><i><font color="Green" title="E124">A13</font></i></a></i>
<br/>.= 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p3">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c16">K1</font>
<b>by </b><i><a class="txt" href="matrix13.html#E6:130_2"><i><font color="Green" title="E121">A10</font></i></a>, <a class="ref" href="matrix13.html#T106" target="_self" title="MATRIX13:th.106">TT106</a>, <a class="txt" href="matrix13.html#E8:130_2"><i><font color="Green" title="E123">A12</font></i></a></i>
;<br/>
<a NAME="E19:130_2"/><b>then </b>
( <font color="Maroon" title="c19">LMK1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p3">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">K1</font>,<font color="Maroon" title="c2">i</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p4">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p3">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">K1</font>,<font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:130"><i><font color="Green" title="E107">A2</font></i></a>, <a class="txt" href="matrix13.html#E7:130_2"><i><font color="Green" title="E122">A11</font></i></a>, <a class="txt" href="matrix13.html#E8:130_2"><i><font color="Green" title="E123">A12</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix13.html#E3:130_2"><i><font color="Green" title="E119">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E20:130_2"/><b>then </b>
( <font color="Maroon" title="c13">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">lmk1i</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font></span>)</span> &amp; <font color="Maroon" title="c17">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">LMK1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font></span>)</span> &amp; <font color="Maroon" title="c22">fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font> &amp; <font color="Maroon" title="c19">LMK1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">lmk1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c22">fk</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E7:130"><i><font color="Green" title="E111">B3</font></i></a>, <a class="txt" href="matrix13.html#E12:130"><i><font color="Green" title="E115">D3</font></i></a>, <a class="txt" href="matrix13.html#E1:130_2"><i><font color="Green" title="E118">A7</font></i></a>, <a class="txt" href="matrix13.html#E4:130_2"><i><font color="Green" title="E120">A9</font></i></a>, <a class="ref" href="matrix13.html#T102" target="_self" title="MATRIX13:th.102">TT102</a></i>;<br/>
<b>hence </b><a NAME="E21:130_2"/>
<font color="Maroon" title="c17">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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#E2:130"><i><font color="Green" title="E107">A2</font></i></a>, <a class="ref" href="fvsum_1.html#T22" title="FVSUM_1:th.22">FVSUM_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:130"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="matrix13.html#E13:130"><i><font color="Green" title="E116">A5</font></i></a>, <a class="txt" href="matrix13.html#E14:130"><i><font color="Green" title="E117">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E16:130"/>
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">FinS2MX</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <span class="p4">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">MX2FinS</a> <font color="Maroon" title="c6">M</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E5:130"><i><font color="Green" title="E109">B1</font></i></a>, <a class="txt" href="matrix13.html#E10:130"><i><font color="Green" title="E113">D1</font></i></a></i>;<br/>


</div>
