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<b>reconsider </b><font color="Maroon" title="c2">i</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">UA</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font> <b>by </b><i><a class="ref" href="endalg.html#T3" target="_self" title="ENDALG:th.3">Th3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">H</font> =  <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K2" title="ENDALG:func.2">UAEndComp</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<font color="Maroon" title="c2">i</font> #);<br/>
<b>set </b><font color="Maroon" title="c4">F</font> =  <a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font>;<br/>
<b>thus </b><a NAME="E2:9_1_1"/>
 <a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<a NAME="E1:9_1_1_1"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 
<b>by </b><i><a class="ref" href="endalg.html#D3" target="_self" title="ENDALG:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E2:9_1_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="txt" href="endalg.html#E8"><i><font color="Green" title="E6">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<a NAME="E3:9_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K2" title="ENDALG:func.2">UAEndComp</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<font color="Maroon" title="c2">i</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 
;<br/>
<a NAME="E4:9_1_1"/>
 <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K2" title="ENDALG:func.2">UAEndComp</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<font color="Maroon" title="c2">i</font> #) is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_1_1_2"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K2" title="ENDALG:func.2">UAEndComp</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<font color="Maroon" title="c2">i</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">h</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c7">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K2" title="ENDALG:func.2">UAEndComp</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<font color="Maroon" title="c2">i</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c10">C</font> = <font color="Maroon" title="c7">h</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font> ;<br/>
<a NAME="E2:9_1_1_2_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">A</font>
 
<b>by </b><i><a class="ref" href="endalg.html#D2" target="_self" title="ENDALG:def.2">Def2</a></i>;<br/>
<a NAME="E3:9_1_1_2_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="ref" href="endalg.html#D2" target="_self" title="ENDALG:def.2">Def2</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h</font> = 
<font color="Maroon" title="c10">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">A</font></span>)</span>
<b>by </b><i><a class="txt" href="endalg.html#E2:9_1_1_2_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="endalg.html#D2" target="_self" title="ENDALG:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">B</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">A</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h</font></span>)</span>
<b>by </b><i><a class="txt" href="endalg.html#E3:9_1_1_2_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="endalg.html#D2" target="_self" title="ENDALG:def.2">Def2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:9_1_1"/>
 <a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>
 <b>by </b><i><a class="txt" href="endalg.html#E3:9_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="endalg.html#D3" target="_self" title="ENDALG:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
