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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b2">L</font> being    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> holds   <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Olive" title="b2">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Olive" title="b2">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b2">L</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> holds   <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">L</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">L</font> be    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c3">L</font></span>)</span></span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">L</font> being    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/> <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">L</font></span>)</span>;<br/>
<a NAME="E1:47"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">L</font> be    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 implies  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:47_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:47_1_1_1"/>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_1_2"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span><br/><div class="add"><a NAME="E2:47_1_1_2"/><span class="kw">then </span>
 not  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:47_1_1_2"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">w</font> = <font color="Maroon" title="c6">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a></span>;<br/><a NAME="E6:47_1_1_2"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1_2"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E4:47_1_1_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="stirl2_1.html#T55" title="STIRL2_1:th.55">STIRL2_1:55</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">Lw</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span><span class="kw"> such that </span><br/><a NAME="E8:47_1_1_2"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_4.html#T37" title="RLVECT_4:th.37">RLVECT_4:37</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">LLw</font> = <font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T54" title="RLVECT_2:th.54">RLVECT_2:54</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47_1_1_2"><span class="lab"><font color="Green" title="E47">A7</font></span></a></span>
;<br/><a NAME="E10:47_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A8</font></span>: 
 not <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T19" title="RLVECT_2:th.19">RLVECT_2:19</a></span>;<br/><a NAME="E11:47_1_1_2"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c8">Lw</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/><a NAME="E12:47_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A10</font></span>: 
(  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c8">Lw</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c8">Lw</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T55" title="RLVECT_2:th.55">RLVECT_2:55</a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></span>;<br/><a NAME="E13:47_1_1_2"/>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1_2"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T40" title="ZFMISC_1:th.40">ZFMISC_1:40</a></span>;<br/><a NAME="E14:47_1_1_2"/><span class="kw">then </span>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:47_1_1_2"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E15:47_1_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_1_1_2"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E6:47_1_1_2"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="nat_1.html#T43" title="NAT_1:th.43">NAT_1:43</a>, <a class="ref" href="zfmisc_1.html#T34" title="ZFMISC_1:th.34">ZFMISC_1:34</a></span>;<br/><a NAME="E16:47_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A11</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E52">A12</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c8">Lw</font> = 
<font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T40" title="RLVECT_2:th.40">RLVECT_2:40</a></span>
<br/>.= 
<font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K4" title="RLVECT_2:func.4">ZeroLC</a> <font color="Maroon" title="c1">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T57" title="RLVECT_2:th.57">RLVECT_2:57</a></span>
<br/>.= 
<font color="Maroon" title="c5">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T41" title="RLVECT_2:th.41">RLVECT_2:41</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A13</font></span>:  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c8">Lw</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_3.html#T1" title="RLVECT_3:th.1">RLVECT_3:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T32" title="RLVECT_2:th.32">RLVECT_2:32</a></span>
;<br/><a NAME="E19:47_1_1_2"/>
<font color="Maroon" title="c2">v</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c8">Lw</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">v</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">w</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:47_1_1_2"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="rltopsp1.html#T8" title="RLTOPSP1:th.8">RLTOPSP1:8</a></span>;<br/><a NAME="E20:47_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c8">Lw</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin1.html#E7"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/><a NAME="E21:47_1_1_2"/><span class="kw">then </span>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c8">Lw</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T16" target="_self" title="RLAFFIN1:th.16">Th16</a></span>;<br/><a NAME="E22:47_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c8">Lw</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A14</font></span>:  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c8">Lw</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c8">Lw</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T32" title="RLVECT_2:th.32">RLVECT_2:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#D1" target="_self" title="RLAFFIN1:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_4.html#T1" title="RLVECT_4:th.1">RLVECT_4:1</a></span>
;<br/><span class="lab"><font color="Green" title="E55">A15</font></span>:  <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c5">L</font> = 
<span class="p1">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c8">Lw</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:47_1_1_2"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="rlaffin1.html#T34" target="_self" title="RLAFFIN1:th.34">Th34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:47_1_1_2"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="rlaffin1.html#T32" target="_self" title="RLAFFIN1:th.32">Th32</a></span>
;<br/><a NAME="E25:47_1_1_2"/>
<font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c8">Lw</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:47_1_1_2"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="rlaffin1.html#T17" target="_self" title="RLAFFIN1:th.17">Th17</a></span>;<br/><span class="kw">hence </span> <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c5">L</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_3.html#T1" title="RLVECT_3:th.1">RLVECT_3:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:47_1_1_2"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E23:47_1_1_2"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c8">Lw</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Lw</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:47_1_1_2"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:47_1_1_2"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:47"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">L</font> be    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c4">L</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c4">L</font></span>)</span></span><br/>

<a NAME="E2:47_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A17</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">V</font>
 
;<br/>
<a NAME="E3:47_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A18</font></span>: 
( <font color="Maroon" title="c4">L</font> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a>  <a href="rlvect_2.html#K4" title="RLVECT_2:func.4">ZeroLC</a> <font color="Maroon" title="c1">V</font> &amp;  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c4">L</font> <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="c1">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T34" title="RLVECT_2:th.34">RLVECT_2:34</a>, <a class="ref" href="rlvect_2.html#D5" title="RLVECT_2:def.5">RLVECT_2:def 5</a></span>;<br/>
<a NAME="E4:47_2"/>
<font color="Maroon" title="c2">v</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_2"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="ref" href="rlaffin1.html#T8" target="_self" title="RLAFFIN1:th.8">Th8</a></span>;<br/>
<a NAME="E5:47_2"/><span class="kw">then </span>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T16" target="_self" title="RLAFFIN1:th.16">Th16</a></span>;<br/>
<span class="kw">hence </span> <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c4">L</font></span>)</span> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T34" title="RLVECT_2:th.34">RLVECT_2:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c4">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_2"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="ref" href="rlaffin1.html#T31" target="_self" title="RLAFFIN1:th.31">Th31</a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:47"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>);<br/></span>
<a NAME="E4:47"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">L</font></span>)</span>]
 
;<br/>
<span class="kw">hence </span><a NAME="E5:47"/>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlaffin1.html#K1" title="RLAFFIN1:func.1">+</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c3">L</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
