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

<b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ,<a href="csspace.html#K6" target="_self">CZeroseq</a> ,<a href="csspace.html#K4" target="_self">l_add</a> ,<a href="csspace.html#K5" target="_self">l_mult</a>  #);<br/>

<b>set </b><font color="Maroon">V</font> =  <a href="clvect_1.html#G1">CLSStruct</a>(# <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ,<a href="csspace.html#K6" target="_self">CZeroseq</a> ,<a href="csspace.html#K4" target="_self">l_add</a> ,<a href="csspace.html#K5" target="_self">l_mult</a>  #);<br/>
<b>reconsider </b><font color="Maroon">w</font> =  <a href="comseq_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span> as   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ,<a href="csspace.html#K6" target="_self">CZeroseq</a> ,<a href="csspace.html#K4" target="_self">l_add</a> ,<a href="csspace.html#K5" target="_self">l_mult</a>  #) <b>by </b><i/>;<br/>
<a NAME="E2:15"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="csspace.html#K6" target="_self">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">E31</font></i>: <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E2:15_1"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <a href="csspace.html#K6" target="_self">CZeroseq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">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">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">s</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9">COMSEQ_1:def 9</a></i>
<br/>.= 
0

<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <a href="csspace.html#K6" target="_self">CZeroseq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><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" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <a href="csspace.html#K6" target="_self">CZeroseq</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>


</div><b>end;</b></div>
<b>then </b><span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span> = 
 <a href="csspace.html#K2" target="_self">seq_id</a> <a href="csspace.html#K6" target="_self">CZeroseq</a> 
<b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>
<br/>.= 
 <a href="csspace.html#K6" target="_self">CZeroseq</a> 
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="csspace.html#K6" target="_self">CZeroseq</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
 ex <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1">CLSStruct</a>(# <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ,<a href="csspace.html#K6" target="_self">CZeroseq</a> ,<a href="csspace.html#K4" target="_self">l_add</a> ,<a href="csspace.html#K5" target="_self">l_mult</a>  #) st <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="clvect_1.html#G1">CLSStruct</a>(# <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ,<a href="csspace.html#K6" target="_self">CZeroseq</a> ,<a href="csspace.html#K4" target="_self">l_add</a> ,<a href="csspace.html#K5" target="_self">l_mult</a>  #)
 ;<br/>


</div>
