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

<b>let </b><font color="Maroon" title="c1">LS</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ) <b>-&gt; </b>  <a href="relset_1.html#M2" title="RELSET_1:mode.2">M5</a>(<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of $1,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of $1</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of $1) = the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of $1;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">LS</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">LS</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 ;<br/>

<a NAME="E3:18"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">LS</font> holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="prvect_2.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:18"/>
<font color="Maroon" title="c1">LS</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a></i>;<br/>

<a NAME="E5:18"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">LS</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prvect_2.html#E2:18"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:18"/>
<font color="Maroon" title="c1">LS</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/>


</div>
