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

<b>reconsider </b><font color="Maroon" title="c2">G1</font> = <font color="Maroon" title="c1">G</font> as   <a href="prvect_2.html#NM2" title="PRVECT_2:NM.2">RealLinearSpace-Sequence</a> ;<br/>
<a NAME="E2:38_1_1"/><i><font color="Green" title="E28">A1</font></i>: 
 <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="prvect_2.html#D13" target="_self" title="PRVECT_2:def.13">Def13</a></i>;<br/>
<a NAME="E3:38_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
 <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</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">v1</font> = <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c6">w1</font> = <font color="Maroon" title="c4">w</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">G1</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E2:38_1_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">w1</font>
 
<b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E3:38_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">v1</font>
 
;<br/>
<b>hence </b><a NAME="E4:38_1_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:38_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
 <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font>, <font color="Maroon" title="c5">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</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">v1</font> = <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c7">w1</font> = <font color="Maroon" title="c4">w</font>, <font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c5">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">G1</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E2:38_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">x1</font>
 
<b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E3:38_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">w1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">x1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:38_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:38_1_1"/><i><font color="Green" title="E31">A4</font></i>: 
 <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3"><b>proof </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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</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">v1</font> = <font color="Maroon" title="c3">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">G1</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E2:38_1_1_3"/>
<font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1: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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">G1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1_3"/>
<font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1: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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:38_1_1"/><i><font color="Green" title="E32">A5</font></i>: 
 <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_4"><b>proof </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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon" title="c4">v1</font> = <font color="Maroon" title="c3">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">G1</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">w</font> =  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">v1</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">w</font>
;<br/>

<a NAME="E3:38_1_1_4"/>
<font color="Maroon" title="c3">v</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="c4">v1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">v1</font>
 
<b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E4:38_1_1_4"/><b>then </b>
<font color="Maroon" title="c3">v</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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">G1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>;<br/>
<b>hence </b><a NAME="E5:38_1_1_4"/>
<font color="Maroon" title="c3">v</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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:38_1_1"/>
 <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">RealLinearSpace-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_5"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E33">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_5_1"><b>now </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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c7">w1</font> = <font color="Maroon" title="c5">w</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">G1</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><a NAME="E2:38_1_1_5_1"/>
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w1</font></span>)</span>
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><a NAME="E3:38_1_1_5_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">w1</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/><b>hence </b><a NAME="E4:38_1_1_5_1"/>
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">w</font></span>)</span>
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E34">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_5_2"><b>now </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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c5">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">G1</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><a NAME="E2:38_1_1_5_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v1</font>
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><a NAME="E3:38_1_1_5_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v1</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/><b>hence </b><a NAME="E4:38_1_1_5_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E35">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_5_3"><b>now </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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c5">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">G1</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><a NAME="E2:38_1_1_5_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v1</font>
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><a NAME="E3:38_1_1_5_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v1</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/><b>hence </b><a NAME="E4:38_1_1_5_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_5_4"><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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c4">v1</font> = <font color="Maroon" title="c3">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c2">G1</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><a NAME="E2:38_1_1_5_4"/>
1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v1</font>
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:38_1_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:38_1_1_5_4"/>
1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:38_1_1_5"/>
 <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">RealLinearSpace-like</a>
 <b>by </b><i><a class="txt" href="prvect_2.html#E1:38_1_1_5"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="prvect_2.html#E2:38_1_1_5"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="prvect_2.html#E3:38_1_1_5"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:38_1_1"/>
(  <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a> &amp;  <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">RealLinearSpace-like</a> &amp;  <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> &amp;  <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> &amp;  <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp;  <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> )
 <b>by </b><i><a class="txt" href="prvect_2.html#E3:38_1_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="prvect_2.html#E4:38_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="prvect_2.html#E5:38_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="prvect_2.html#E6:38_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="prvect_2.html#E29"><i><font color="Green" title="E27">Lm7</font></i></a></i>;<br/>


</div>
