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

<b>reconsider </b><font color="Maroon">i</font> =  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> as    <a href="autalg_1.html#M2">Element</a> of  <a href="endalg.html#K4" target="_self">MSAEnd</a> <font color="Maroon">U1</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" target="_self">MSAEnd</a> <font color="Maroon">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" target="_self">MSAEndComp</a> <font color="Maroon">U1</font></span>)</span>,<font color="Maroon">i</font> #);<br/>
<b>thus </b><a NAME="E2:21_1_1"/>
 <a href="endalg.html#K6" target="_self">MSAEndMonoid</a> <font color="Maroon">U1</font> is <a href="group_1.html#V2">unital</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="endalg.html#K6" target="_self">MSAEndMonoid</a> <font color="Maroon">U1</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D2">GROUP_1:def 2</a><br/>

<b>thus </b><a NAME="E2:21_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="endalg.html#K6" target="_self">MSAEndMonoid</a> <font color="Maroon">U1</font></span>)</span> holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i/>;<br/>


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

<a NAME="E3:21_1_1"/>
 <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" target="_self">MSAEnd</a> <font color="Maroon">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" target="_self">MSAEndComp</a> <font color="Maroon">U1</font></span>)</span>,<font color="Maroon">i</font> #) is <a href="group_1.html#V4">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:21_1_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" target="_self">MSAEnd</a> <font color="Maroon">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" target="_self">MSAEndComp</a> <font color="Maroon">U1</font></span>)</span>,<font color="Maroon">i</font> #)<br/> for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="autalg_1.html#M2">Element</a> of  <a href="endalg.html#K4" target="_self">MSAEnd</a> <font color="Maroon">U1</font>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> holds <br/><font color="Olive">f</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="msualg_3.html#K3">**</a> <font color="Olive">A</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:21_1_1_2"/>
for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" target="_self">MSAEnd</a> <font color="Maroon">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" target="_self">MSAEndComp</a> <font color="Maroon">U1</font></span>)</span>,<font color="Maroon">i</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</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="21_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" target="_self">MSAEnd</a> <font color="Maroon">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" target="_self">MSAEndComp</a> <font color="Maroon">U1</font></span>)</span>,<font color="Maroon">i</font> #), <font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" target="_self">MSAEnd</a> <font color="Maroon">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" target="_self">MSAEndComp</a> <font color="Maroon">U1</font></span>)</span>,<font color="Maroon">i</font> #), <font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" target="_self">MSAEnd</a> <font color="Maroon">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" target="_self">MSAEndComp</a> <font color="Maroon">U1</font></span>)</span>,<font color="Maroon">i</font> #);<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">f</font>, <font color="Maroon">B</font> = <font color="Maroon">g</font>, <font color="Maroon">C</font> = <font color="Maroon">h</font> as    <a href="autalg_1.html#M2">Element</a> of  <a href="endalg.html#K4" target="_self">MSAEnd</a> <font color="Maroon">U1</font> ;<br/>
<a NAME="E2:21_1_1_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:21_1_1_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<font color="Maroon">C</font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">B</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_1_1"/>
 <a href="endalg.html#K6" target="_self">MSAEndMonoid</a> <font color="Maroon">U1</font> is <a href="group_1.html#V4">associative</a>
 <b>by </b><i/>;<br/>


</div>
