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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <a href="rsspace.html#K1" target="_self">the_set_of_RealSequences</a> ,<a href="rsspace.html#K6" target="_self">Zeroseq</a> ,<a href="rsspace.html#K4" target="_self">l_add</a> ,<a href="rsspace.html#K5" target="_self">l_mult</a>  #);<br/>



<a NAME="E1:17"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rsspace.html#E1:17_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rsspace.html#E1:17_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rsspace.html#E1:17_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E3:17_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rsspace.html#E1:17_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T8">SEQ_1:8</a></i>;<br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3" target="_self">Th3</a></i>
;<br/>
<b>hence </b><a NAME="E3:17"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rsspace.html#E1:17"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rsspace.html#T5" target="_self">Th5</a></i>;<br/>


</div>
