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

<b>let </b><font color="Maroon" title="c1">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a> ,<a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> ,<a href="rsspace.html#K4" title="RSSPACE:func.4">l_add</a> ,<a href="rsspace.html#K5" title="RSSPACE:func.5">l_mult</a>  #);<br/>

<b>set </b><font color="Maroon" title="c2">V</font> =  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a> ,<a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> ,<a href="rsspace.html#K4" title="RSSPACE:func.4">l_add</a> ,<a href="rsspace.html#K5" title="RSSPACE:func.5">l_mult</a>  #);<br/>
<a NAME="E1:14"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c1">v</font> <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> <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a> ,<a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> ,<a href="rsspace.html#K4" title="RSSPACE:func.4">l_add</a> ,<a href="rsspace.html#K5" title="RSSPACE:func.5">l_mult</a>  #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rsspace.html#T4" target="_self" title="RSSPACE:th.4">Th4</a></i>;<br/>
<a NAME="E2:14"/>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">s</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">s</font></span>)</span>
<b>by </b><i><a class="txt" href="rsspace.html#E1:14_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0
<b>by </b><i><a class="txt" href="rsspace.html#E1:14_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="rsspace.html#D6" target="_self" title="RSSPACE:def.6">Def6</a></i>
;<br/>
<b>hence </b><a NAME="E3:14_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">s</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><font color="Maroon" title="c1">v</font> <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> <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a> ,<a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> ,<a href="rsspace.html#K4" title="RSSPACE:func.4">l_add</a> ,<a href="rsspace.html#K5" title="RSSPACE:func.5">l_mult</a>  #)</span>)</span> = 
 <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font>
<b>by </b><i><a class="txt" href="rsspace.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">SEQ_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c1">v</font>
<b>by </b><i><a class="ref" href="rsspace.html#D2" target="_self" title="RSSPACE:def.2">Def2</a></i>
;<br/><br/>


</div>
