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

<b>thus </b><a NAME="E1:28_1_1"/>
 <a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font> is <a href="bhsp_1.html#V2">RealUnitarySpace-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font>, <font color="Maroon">z1</font> = <font color="Maroon">z</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x1</font>, <font color="Maroon">y2</font> = <font color="Maroon">y1</font>, <font color="Maroon">z2</font> = <font color="Maroon">z1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><a NAME="E3:28_1_1_1_1"/><i><font color="Green">E33</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><i><font color="Green">E34</font></i>: <font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E53</font></i>: <font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E61</font></i>: <font color="Maroon">y</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E7:28_1_1_1_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1_1_1_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font></span>)</span>
 ;<br/>

<a NAME="E2:28_1_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T86">RVSUM_1:86</a></i>;<br/>
<b>hence </b><a NAME="E3:28_1_1_1_1_4"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:28_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a></i>;<br/><b>thus </b><a NAME="E9:28_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_1_1_5_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> )
 ;<br/><a NAME="E2:28_1_1_1_1_5_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x2</font> holds <br/><font color="Maroon">x2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1_1_1_5_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E2:28_1_1_1_1_5_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K4">dom</a> <a href="binop_2.html#K35">multreal</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:28_1_1_1_1_5_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">x1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="binop_2.html#K35">multreal</a> 
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E4:28_1_1_1_1_5_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="funcop_1.html#K3">.:</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T84">FUNCOP_1:84</a></i>;<br/>
<a NAME="E5:28_1_1_1_1_5_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="rvsum_1.html#D9">RVSUM_1:def 9</a></i>;<br/>
<a NAME="E6:28_1_1_1_1_5_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 
<b>by </b><i>, , , <a class="ref" href="rvsum_1.html#T115">RVSUM_1:115</a></i>;<br/>
<a NAME="E7:28_1_1_1_1_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E8:28_1_1_1_1_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T86">RVSUM_1:86</a></i>;<br/>
<a NAME="E9:28_1_1_1_1_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E10:28_1_1_1_1_5_1_1"/>
<font color="Maroon">x2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:28_1_1_1_1_5_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0
 <b>by </b><i><a class="ref" href="rfunct_4.html#T4">RFUNCT_4:4</a></i>;<br/><a NAME="E4:28_1_1_1_1_5_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>;<br/><b>hence </b><a NAME="E5:28_1_1_1_1_5_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:28_1_1_1_1_5"/>
( <font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E3:28_1_1_1_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<b>then </b><font color="Maroon">x</font> = 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
;<br/>
<b>then </b> <a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">x2</font> = 
0 <a href="rvsum_1.html#K10">*</a> <font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T92">RVSUM_1:92</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0
<b>by </b><i><a class="ref" href="rvsum_1.html#T75">RVSUM_1:75</a></i>
;<br/>
<b>hence </b><a NAME="E6:28_1_1_1_1_5"/>
<font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T111">RVSUM_1:111</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">y1</font>,<font color="Maroon">x1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<font color="Maroon">z1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">y1</font></span>)</span>,<font color="Maroon">z1</font>
<b>by </b><i/>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">y1</font></span>)</span>,<font color="Maroon">z1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid_2.html#T9">EUCLID_2:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T119">RVSUM_1:119</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
;<br/><b>reconsider </b><font color="Maroon">ax1</font> = <font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font>, <font color="Maroon">ay1</font> = <font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y1</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> ;<br/><b>reconsider </b><font color="Maroon">ax2</font> = <font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">x2</font>, <font color="Maroon">ay2</font> = <font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">y2</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E14:28_1_1_1_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">ax</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><a NAME="E16:28_1_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><a NAME="E17:28_1_1_1_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">ax</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1_1_1_8"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T132">FINSEQ_2:132</a></i>;<br/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">x1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:28_1_1_1_1_8"/>
<font color="Maroon">ax</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span>,<font color="Maroon">y1</font>
<b>by </b><i>, <a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span>,<font color="Maroon">y1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T94">RVSUM_1:94</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:28_1_1_1"/>
 <a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font> is <a href="bhsp_1.html#V2">RealUnitarySpace-like</a>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:28_1_1"/>
 <a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V7">RealLinearSpace-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_1_1_2"/>
for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font>, <font color="Maroon">w2</font> = <font color="Maroon">w1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<i><font color="Green">E77</font></i>: <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<font color="Maroon">w1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T73">RVSUM_1:73</a></i>
;<br/>
<a NAME="E4:28_1_1_2_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">av</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E6:28_1_1_2_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E7:28_1_1_2_1"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">av</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T132">FINSEQ_2:132</a></i>;<br/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E4:28_1_1_2_1_2"/>
<font color="Maroon">av</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:28_1_1_2_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">aw</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E10:28_1_1_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">w1</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E11:28_1_1_2_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">aw</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">w1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1_2_1_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T132">FINSEQ_2:132</a></i>;<br/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">w1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">w1</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:28_1_1_2_1_3"/>
<font color="Maroon">aw</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">w1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:28_1_1_2_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">w2</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E14:28_1_1_2_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:28_1_1_2"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="binop_2.html#K9">+</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<i><font color="Green">E90</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">b</font></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T72">RVSUM_1:72</a></i>
;<br/>
<a NAME="E4:28_1_1_2_2"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">av</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E6:28_1_1_2_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E7:28_1_1_2_2"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">av</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1_2_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">a</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T132">FINSEQ_2:132</a></i>;<br/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:28_1_1_2_2_2"/>
<font color="Maroon">av</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:28_1_1_2_2"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">bv</font> = <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E10:28_1_1_2_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E11:28_1_1_2_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">bv</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1_2_2_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">b</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T132">FINSEQ_2:132</a></i>;<br/>
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:28_1_1_2_2_3"/>
<font color="Maroon">bv</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:28_1_1_2_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E14:28_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:28_1_1_2"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="binop_2.html#K11">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E3:28_1_1_2_3"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">bv</font> = <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E5:28_1_1_2_3"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E6:28_1_1_2_3"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">bv</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1_2_3_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">b</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T132">FINSEQ_2:132</a></i>;<br/>
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:28_1_1_2_3_1"/>
<font color="Maroon">bv</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:28_1_1_2_3"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>;<br/>
<i><font color="Green">E114</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b</font></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71">RVSUM_1:71</a></i>
;<br/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E10:28_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<a NAME="E4:28_1_1_2"/>
for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> holds  1 <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> 1,<font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
1 <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T74">RVSUM_1:74</a></i>
;<br/>
<b>hence </b><a NAME="E4:28_1_1_2_4"/>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:28_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> holds  1 <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:28_1_1"/>
 <a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V3">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">w1</font>,<font color="Maroon">v1</font>
<b>by </b><i><a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>

<a NAME="E4:28_1_1"/>
for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font>, <font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">u2</font> = <font color="Maroon">u1</font>, <font color="Maroon">v2</font> = <font color="Maroon">v1</font>, <font color="Maroon">w2</font> = <font color="Maroon">w1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<i><font color="Green">E115</font></i>: <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">w1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font></span>)</span>,<font color="Maroon">w1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">v1</font></span>)</span>,<font color="Maroon">w1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">v1</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">w2</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">u2</font> <a href="rvsum_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="rvsum_1.html#K4">+</a> <font color="Maroon">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T32">RVSUM_1:32</a></i>
;<br/>
<font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">u1</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">u2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">v2</font>,<font color="Maroon">w2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">u2</font>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:28_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:28_1_1"/>
 <a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V4">add-associative</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>

<a NAME="E6:28_1_1"/>
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:28_1_1_5"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:28_1_1"/>
 <a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V5">right_zeroed</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>;<br/>

<a NAME="E8:28_1_1"/>
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>  ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> st <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">w</font> =  <a href="euclid.html#K6">-</a> <font color="Maroon">v1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="euclidlp.html#T7">EUCLIDLP:7</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E4:28_1_1_6"/>
 ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span> st <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:28_1_1"/>
 <a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V6">right_complementable</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>


</div>
