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

<b>let </b><font color="Maroon">B</font> be   <a href="vectsp_1.html#NM1">AbGroup</a>;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #);<br/>
<a NAME="E1:80"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #)<br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> holds <br/><font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>
<a NAME="E2:80"/><i><font color="Green">E53</font></i>: 
(  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #) is <a href="group_1.html#V4">associative</a> &amp;  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #) is <a href="group_1.html#V3">Group-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:80_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Olive">c</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #), <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #);<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font>, <font color="Maroon">z</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> ;<br/>
<a NAME="E2:80_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/>


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

<b>reconsider </b><font color="Maroon">e</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">B</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #) ;<br/>
<b>take </b>
<font color="Maroon">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #);<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> ;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> = 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>

<b>thus </b><font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> = 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>

<b>reconsider </b><font color="Maroon">b</font> =  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #) ;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> = 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
;<br/>

<b>thus </b><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> = 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #);<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> ;<br/><b>thus </b><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> = 
<font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:80"/>
 <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> #) is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D11" target="_self">Def11</a>, <a class="ref" href="group_1.html#D16">GROUP_1:def 16</a></i>;<br/>


</div>
