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

<b>let </b><font color="Maroon" title="c1">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> ,<a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> ,<a href="csspace.html#K4" title="CSSPACE:func.4">l_add</a> ,<a href="csspace.html#K5" title="CSSPACE:func.5">l_mult</a>  #);<br/>

<b>set </b><font color="Maroon" title="c2">V</font> =  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> ,<a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> ,<a href="csspace.html#K4" title="CSSPACE:func.4">l_add</a> ,<a href="csspace.html#K5" title="CSSPACE:func.5">l_mult</a>  #);<br/>
<b>reconsider </b><font color="Maroon" title="c3">w</font> =  <a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> ,<a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> ,<a href="csspace.html#K4" title="CSSPACE:func.4">l_add</a> ,<a href="csspace.html#K5" title="CSSPACE:func.5">l_mult</a>  #) <b>by </b><i><a class="ref" href="csspace.html#D1" target="_self" title="CSSPACE:def.1">Def1</a></i>;<br/>
<a NAME="E2:15"/>
<font color="Maroon" title="c1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E16">A1</font></i>: <font color="Maroon" title="c1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c3">w</font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">w</font></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T4" target="_self" title="CSSPACE:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p1">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#T3" target="_self" title="CSSPACE:th.3">Th3</a></i>
;<br/>
<a NAME="E2:15_1"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p2">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</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="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p2">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">s</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4" title="COMSEQ_1:def.4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9" title="COMSEQ_1:def.9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">s</font>
<b>by </b><i><a class="ref" href="csspace.html#D6" target="_self" title="CSSPACE:def.6">Def6</a></i>
;<br/>
<b>hence </b><a NAME="E2:15_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p2">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">s</font>
 ;<br/>


</div><b>end;</b></div>
<b>then </b><span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <span class="p1">(<span class="default"><a href="comseq_1.html#K5" title="COMSEQ_1:func.5">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">v</font></span>)</span></span>)</span> = 
 <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> 
<b>by </b><i><a class="ref" href="comseq_1.html#T6" title="COMSEQ_1:th.6">COMSEQ_1:6</a></i>
<br/>.= 
 <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> 
<b>by </b><i><a class="ref" href="csspace.html#D2" target="_self" title="CSSPACE:def.2">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon" title="c1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> 
 <b>by </b><i><a class="txt" href="csspace.html#E1:15_1"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
 ex <font color="Olive" title="b1">w</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> ,<a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> ,<a href="csspace.html#K4" title="CSSPACE:func.4">l_add</a> ,<a href="csspace.html#K5" title="CSSPACE:func.5">l_mult</a>  #) st <font color="Maroon" title="c1">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> ,<a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> ,<a href="csspace.html#K4" title="CSSPACE:func.4">l_add</a> ,<a href="csspace.html#K5" title="CSSPACE:func.5">l_mult</a>  #)
 ;<br/>


</div>
