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

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v3</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="rlvect_2.html#M2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K8">}</a></span>;<br/>





<b>assume </b><a NAME="E1:42"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v3</font> &amp; <font color="Maroon">v3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font> )
 ;<br/>

<a NAME="E2:42"/><i><font color="Green">E32</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K8">}</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1"/>
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v1</font></span><a href="domain_1.html#K6">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v2</font></span><a href="domain_1.html#K6">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v3</font></span><a href="domain_1.html#K6">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K7">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K7">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K8">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="zfmisc_1.html#T142">ZFMISC_1:142</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>suppose </b></a><a NAME="E1:42_1_1"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:42_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">L</font> <a href="rlvect_2.html#R1">=</a>  <a href="rlvect_2.html#K4">ZeroLC</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D7">RLVECT_2:def 7</a></i>;<br/><b>then </b> <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_2.html#T48">RLVECT_2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_2.html#T30">RLVECT_2:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_2.html#T30">RLVECT_2:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_2.html#T30">RLVECT_2:30</a></i>
;<br/><b>hence </b><a NAME="E4:42_1_1"/>
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>suppose </b></a><a NAME="E1:42_1_2"/><i><font color="Green">E37</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v1</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v1</font></span><a href="domain_1.html#K6">}</a></span> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><a NAME="E3:42_1_2"/><i><font color="Green">E38</font></i>: 
( not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/> <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font>
<b>by </b><i><a class="ref" href="rlvect_2.html#T50">RLVECT_2:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>
;<br/><b>hence </b><a NAME="E5:42_1_2"/>
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_3"><b>suppose </b></a><a NAME="E1:42_1_3"/><i><font color="Green">E46</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v2</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v2</font></span><a href="domain_1.html#K6">}</a></span> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><a NAME="E3:42_1_3"/><i><font color="Green">E47</font></i>: 
( not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/> <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="rlvect_2.html#T50">RLVECT_2:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>
;<br/><b>hence </b><a NAME="E5:42_1_3"/>
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4"><b>suppose </b></a><a NAME="E1:42_1_4"/><i><font color="Green">E51</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v3</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v3</font></span><a href="domain_1.html#K6">}</a></span> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><a NAME="E3:42_1_4"/><i><font color="Green">E52</font></i>: 
( not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/> <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font>
<b>by </b><i><a class="ref" href="rlvect_2.html#T50">RLVECT_2:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>
;<br/><b>hence </b><a NAME="E5:42_1_4"/>
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_5"><b>suppose </b></a><a NAME="E1:42_1_5"/><i><font color="Green">E53</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><div class="add"><a NAME="E2:42_1_5"/><b>then </b><i><font color="Green">E55</font></i>: 
not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/> <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_2.html#T54">RLVECT_2:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/><b>hence </b><a NAME="E4:42_1_5"/>
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_6"><b>suppose </b></a><a NAME="E1:42_1_6"/><i><font color="Green">E60</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><div class="add"><a NAME="E2:42_1_6"/><b>then </b><i><font color="Green">E61</font></i>: 
not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/> <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_2.html#T54">RLVECT_2:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/><b>hence </b><a NAME="E4:42_1_6"/>
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_7"><b>suppose </b></a><a NAME="E1:42_1_7"/><i><font color="Green">E62</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><div class="add"><a NAME="E2:42_1_7"/><b>then </b><i><font color="Green">E63</font></i>: 
not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/> <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_2.html#T54">RLVECT_2:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/><b>hence </b><a NAME="E4:42_1_7"/>
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_8"><b>suppose </b></a><a NAME="E1:42_1_8"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K8">}</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:42_1_8"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K8">}</a></span> )
 <b>and </b><br/><a NAME="E4:42_1_8"/><i><font color="Green">E104</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D10">RLVECT_2:def 10</a></i>;<br/><a NAME="E5:42_1_8"/>
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v3</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v2</font>,<font color="Maroon">v1</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v2</font>,<font color="Maroon">v3</font>,<font color="Maroon">v1</font></span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v3</font>,<font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v3</font>,<font color="Maroon">v2</font>,<font color="Maroon">v1</font></span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <b>by </b><i>, , </i>;<br/><a NAME="E6:42_1_8"/><b>then </b>
( <font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="rlvect_2.html#T44">RLVECT_2:44</a></i>;<br/><a NAME="E7:42_1_8"/><b>then </b>
(  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span> or  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span></span>)</span> or  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T63">RLVECT_1:63</a></i>;<br/><b>hence </b><a NAME="E8:42_1_8"/>
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
