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

<b>set </b><font color="Maroon">S</font> =  <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #);<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #), <font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #), <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #);<br/><a NAME="E1:77_1_1_1"/><i><font color="Green">E46</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">v</font>, <font color="Maroon">y</font> = <font color="Maroon">u</font>, <font color="Maroon">z</font> = <font color="Maroon">w</font>, <font color="Maroon">yx</font> = <font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font>, <font color="Maroon">xz</font> = <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font>, <font color="Maroon">ax</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>, <font color="Maroon">az</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>, <font color="Maroon">bx</font> = <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font> ;<br/><a NAME="E3:77_1_1_1"/>
(  <a href="rlvect_2.html#K12" target="_self">@</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp;  <a href="rlvect_2.html#K12" target="_self">@</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp;  <a href="rlvect_2.html#K12" target="_self">@</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp;  <a href="rlvect_2.html#K12" target="_self">@</a> <font color="Maroon">yx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font> &amp;  <a href="rlvect_2.html#K12" target="_self">@</a> <font color="Maroon">xz</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> &amp;  <a href="rlvect_2.html#K12" target="_self">@</a> <font color="Maroon">ax</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> &amp;  <a href="rlvect_2.html#K12" target="_self">@</a> <font color="Maroon">az</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> &amp;  <a href="rlvect_2.html#K12" target="_self">@</a> <font color="Maroon">bx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">K</font> = <font color="Maroon">v</font>, <font color="Maroon">L</font> = <font color="Maroon">u</font>, <font color="Maroon">M</font> = <font color="Maroon">w</font>, <font color="Maroon">LK</font> = <font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font>, <font color="Maroon">KM</font> = <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font>, <font color="Maroon">aK</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>, <font color="Maroon">aM</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>, <font color="Maroon">bK</font> = <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> as    <a href="rlvect_2.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font> ;<br/><div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #), <font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #);<br/><b>let </b><font color="Maroon">K</font> be    <a href="rlvect_2.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="rlvect_2.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:77_1_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> )
 ;<br/><a NAME="E2:77_1_1_1_1"/><i><font color="Green">E54</font></i>: 
(  <a href="rlvect_2.html#K12" target="_self">@</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <font color="Maroon">K</font></span>)</span> <a href="rlvect_2.html#R1" target="_self">=</a> <font color="Maroon">K</font> &amp;  <a href="rlvect_2.html#K12" target="_self">@</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <font color="Maroon">L</font></span>)</span> <a href="rlvect_2.html#R1" target="_self">=</a> <font color="Maroon">L</font> )
 ;<br/><b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <font color="Maroon">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">K</font> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">L</font>
<b>by </b><i><a class="txt" href="rlvect_2.html#E5:77_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rlvect_2.html#D1" target="_self">Def1</a></i>
;<br/></div><b>end;</b></div><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #);<br/><b>let </b><font color="Maroon">L</font> be    <a href="rlvect_2.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:77_1_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E2:77_1_1_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="rlvect_2.html#K12" target="_self">@</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <font color="Maroon">L</font></span>)</span> <a href="rlvect_2.html#R1" target="_self">=</a> <font color="Maroon">L</font>
 ;<br/><b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="rlvect_2.html#E5:77_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">L</font>
<b>by </b><i>, <a class="ref" href="rlvect_2.html#T3" target="_self">Th3</a></i>
;<br/></div><b>end;</b></div><b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">K</font> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">M</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">M</font> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">K</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font>
<b>by </b><i/>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">LK</font> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">M</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">K</font></span>)</span> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">M</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">L</font> <a href="rlvect_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">L</font> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">KM</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
;<br/><b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #)</span>)</span> = 
<font color="Maroon">K</font> <a href="rlvect_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#T4" target="_self">Th4</a>, </i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i/>
;<br/><a NAME="E10:77_1_1_1"/>
 <a href="rlvect_2.html#K9" target="_self">-</a> <font color="Maroon">K</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #)
 <b>by </b><i><a class="txt" href="rlvect_2.html#E1:77_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E11:77_1_1_1"/><b>then </b>
 <a href="rlvect_2.html#K9" target="_self">-</a> <font color="Maroon">K</font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E12:77_1_1_1"/><b>then </b>
 <a href="rlvect_2.html#K9" target="_self">-</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K1" target="_self">vector</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #),<span class="p1">(<span class="default"><a href="rlvect_2.html#K9" target="_self">-</a> <font color="Maroon">K</font></span>)</span>
 <b>by </b><i/>;<br/><b>then </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K1" target="_self">vector</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #),<span class="p2">(<span class="default"><a href="rlvect_2.html#K9" target="_self">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> = 
<font color="Maroon">K</font> <a href="rlvect_2.html#K10" target="_self">-</a> <font color="Maroon">K</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #)
<b>by </b><i><a class="ref" href="rlvect_2.html#T4" target="_self">Th4</a>, </i>
;<br/><b>hence </b><a NAME="E14:77_1_1_1"/>
 ex <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) st <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #)
 ;<br/><b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span> = 
<font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">KM</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">K</font></span>)</span> <a href="rlvect_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">aK</font> <a href="rlvect_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">aK</font> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">aM</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">K</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">K</font></span>)</span> <a href="rlvect_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">aK</font> <a href="rlvect_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">aK</font> <a href="rlvect_2.html#K7" target="_self">+</a> <font color="Maroon">bK</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i/>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">K</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">bK</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i/>
;<br/><b>thus </b>1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
1 <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">K</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<a NAME="E2:77_1_1"/><b>then </b>
( ( for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font> ) &amp; ( for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span> ) &amp; ( for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #)</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> ) &amp; ( for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #)  ex <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) st <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) ) &amp; ( for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) holds  <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K3">+</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> ) &amp; ( for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) holds  1 <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:77_1_1"/>
 <a href="rlvect_1.html#G2">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="rlvect_2.html#K11" target="_self">LinComb</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K13" target="_self">@</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K14" target="_self">LCAdd</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K15" target="_self">LCMult</a> <font color="Maroon">V</font></span>)</span> #) is   <a href="rlvect_1.html#NM2">RealLinearSpace</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T7">RLVECT_1:7</a></i>;<br/>


</div>
