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

<b>set </b><font color="Maroon" title="c2">S</font> =  <a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font>;<br/>
<div><i><font color="Green" title="E44">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">K</font>, <font color="Maroon" title="c6">L</font> be    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font>;<br/><b>assume </b><a NAME="E1:76_1_1_1"/><i><font color="Green" title="E45">A2</font></i>: 
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> )
 ;<br/><a NAME="E2:76_1_1_1"/>
(  <a href="rlvect_2.html#K12" title="RLVECT_2:func.12">@</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K13" title="RLVECT_2:func.13">@</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <font color="Maroon" title="c5">K</font> &amp;  <a href="rlvect_2.html#K12" title="RLVECT_2:func.12">@</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K13" title="RLVECT_2:func.13">@</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <font color="Maroon" title="c6">L</font> )
 ;<br/><b>hence </b><a NAME="E3:76_1_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">K</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="rlvect_2.html#D19" target="_self" title="RLVECT_2:def.19">Def19</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E2:76_1_1"/>
 <a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">K</font> = <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c6">L</font> = <font color="Maroon" title="c4">v</font> as    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">v</font> = 
<font color="Maroon" title="c6">L</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c5">K</font>
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E44">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">u</font>
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E44">A1</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:76_1_1"/>
 <a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">L</font> = <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c7">K</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c8">M</font> = <font color="Maroon" title="c5">w</font> as    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></i>;<br/>
<a NAME="E2:76_1_1_3"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">K</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c8">M</font>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>
<a NAME="E3:76_1_1_3"/>
<font color="Maroon" title="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c7">K</font>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c8">M</font>
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E44">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">L</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">K</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c8">M</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#T61" target="_self" title="RLVECT_2:th.61">Th61</a></i>
<br/>.= 
<font color="Maroon" title="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="rlvect_2.html#E2:76_1_1_3"><i><font color="Green" title="E45">A3</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:76_1_1"/>
 <a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon" title="c4">K</font> = <font color="Maroon" title="c3">v</font> as    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">K</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K4" title="RLVECT_2:func.4">ZeroLC</a> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E44">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">v</font>
<b>by </b><i><a class="ref" href="rlvect_2.html#T62" target="_self" title="RLVECT_2:th.62">Th62</a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:76_1_1"/>
 <a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a><br/>

<b>reconsider </b><font color="Maroon" title="c4">K</font> = <font color="Maroon" title="c3">v</font> as    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></i>;<br/>
<a NAME="E2:76_1_1_5"/>
 <a href="rlvect_2.html#K9" title="RLVECT_2:func.9">-</a> <font color="Maroon" title="c4">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></i>;<br/>
<a NAME="E3:76_1_1_5"/><b>then </b>
 <a href="rlvect_2.html#K9" title="RLVECT_2:func.9">-</a> <font color="Maroon" title="c4">K</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E4:76_1_1_5"/><b>then </b>
 <a href="rlvect_2.html#K9" title="RLVECT_2:func.9">-</a> <font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_2.html#K9" title="RLVECT_2:func.9">-</a> <font color="Maroon" title="c4">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D1" target="_self" title="RLVECT_2:def.1">Def1</a></i>;<br/>
<b>then </b><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_2.html#K9" title="RLVECT_2:func.9">-</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">K</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c4">K</font>
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E44">A1</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#T82" target="_self" title="RLVECT_2:th.82">Th82</a></i>
;<br/>
<b>hence </b><a NAME="E6:76_1_1_5"/>
 ex <font color="Olive" title="b1">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a><br/>


</div><b>end;</b></div>

<div><i><font color="Green" title="E45">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">L</font> be    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:76_1_1_6"/><i><font color="Green" title="E46">A5</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font>
 ;<br/><a NAME="E2:76_1_1_6"/>
 <a href="rlvect_2.html#K12" title="RLVECT_2:func.12">@</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K13" title="RLVECT_2:func.13">@</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <font color="Maroon" title="c4">L</font>
 ;<br/><b>hence </b><a NAME="E3:76_1_1_6"/>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c4">L</font>
 <b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1_6"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="rlvect_2.html#D20" target="_self" title="RLVECT_2:def.20">Def20</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E7:76_1_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">w</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon" title="c6">K</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c7">M</font> = <font color="Maroon" title="c5">w</font> as    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></i>;<br/>
<a NAME="E2:76_1_1_7"/><i><font color="Green" title="E46">A6</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c7">M</font> )
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E6:76_1_1"><i><font color="Green" title="E45">A4</font></i></a></i>;<br/>
<a NAME="E3:76_1_1_7"/>
<font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">K</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c7">M</font>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>
<b>hence </b><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <font color="Maroon" title="c7">M</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_2.html#E6:76_1_1"><i><font color="Green" title="E45">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c6">K</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c7">M</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#T69" target="_self" title="RLVECT_2:th.69">Th69</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">w</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="rlvect_2.html#E2:76_1_1_7"><i><font color="Green" title="E46">A6</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E8:76_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c5">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>





<b>reconsider </b><font color="Maroon" title="c6">K</font> = <font color="Maroon" title="c5">v</font> as    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></i>;<br/>
<a NAME="E2:76_1_1_8"/><i><font color="Green" title="E46">A7</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c6">K</font> )
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E6:76_1_1"><i><font color="Green" title="E45">A4</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c6">K</font>
<b>by </b><i><a class="txt" href="rlvect_2.html#E6:76_1_1"><i><font color="Green" title="E45">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c6">K</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c6">K</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#T68" target="_self" title="RLVECT_2:th.68">Th68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_2.html#E1:76_1_1"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="rlvect_2.html#E2:76_1_1_8"><i><font color="Green" title="E46">A7</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E9:76_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c5">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>





<b>reconsider </b><font color="Maroon" title="c6">K</font> = <font color="Maroon" title="c5">v</font> as    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></i>;<br/>
<a NAME="E2:76_1_1_9"/><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c6">K</font>
 
<b>by </b><i><a class="txt" href="rlvect_2.html#E6:76_1_1"><i><font color="Green" title="E45">A4</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c6">K</font>
<b>by </b><i><a class="txt" href="rlvect_2.html#E6:76_1_1"><i><font color="Green" title="E45">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c6">K</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_2.html#T70" target="_self" title="RLVECT_2:th.70">Th70</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_2.html#E6:76_1_1"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="rlvect_2.html#E2:76_1_1_9"><i><font color="Green" title="E46">A8</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rlvect_2.html#K16" title="RLVECT_2:func.16">LC_RLSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c4">K</font> = <font color="Maroon" title="c3">v</font> as    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_2.html#D16" target="_self" title="RLVECT_2:def.16">Def16</a></i>;<br/>
<b>thus </b>1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font> = 
1 <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c4">K</font>
<b>by </b><i><a class="txt" href="rlvect_2.html#E6:76_1_1"><i><font color="Green" title="E45">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">v</font>
<b>by </b><i><a class="ref" href="rlvect_2.html#T71" target="_self" title="RLVECT_2:th.71">Th71</a></i>
;<br/>


</div>
