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<b>let </b><font color="Maroon">A</font> be  <a href="group_1.html#V7" target="_self">commutative</a> <a href="group_1.html#NM1">Group</a>;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #);<br/>
<b>thus </b><a NAME="E1:88"/>
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #) is <a href="rlvect_1.html#V3">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_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="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><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">A</font> ;<br/>
<a NAME="E2:88_1"/><i><font color="Green">E40</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="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #)<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">A</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="rlvect_1.html#K2">+</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="group_1.html#K10" target="_self">*</a> <font color="Olive">y</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>
<b>hence </b><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> = 
<font color="Maroon">x</font> <a href="group_1.html#K10" target="_self">*</a> <font color="Maroon">y</font>

<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a</font>
<b>by </b><i/>
;<br/><br/>


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

<a NAME="E2:88"/><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="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #)<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">A</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="rlvect_1.html#K2">+</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="group_1.html#K10" target="_self">*</a> <font color="Olive">y</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #), <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #);<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">A</font> ;<br/><a NAME="E2:88_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K10" target="_self">*</a> <font color="Maroon">y</font> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K10" target="_self">*</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="rlvect_1.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K10" target="_self">*</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K10" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K10" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #);<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">A</font> ;<br/><b>thus </b><font color="Maroon">a</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#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #)</span>)</span> = 
<font color="Maroon">x</font> <a href="group_1.html#K10" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><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">A</font> ;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="group_1.html#K4" target="_self">inverse_op</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #) ;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> = 
<font color="Maroon">x</font> <a href="group_1.html#K10" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" target="_self">inverse_op</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K10" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="group_1.html#U1" target="_self">mult</a> of <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">A</font></span>)</span> #)
<b>by </b><i/>
;<br/>


</div>
