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

<a NAME="E1:30_1_1"/>
( <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font> is <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_1_1_1"/>
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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font></span>)</span> holds  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>
  <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"> <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font> is <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E2:30_1_1_1_1"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:30_1_1_1_1"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c7">c</font> )
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_6.html#K6" title="GROUP_6:func.6">@</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K6" title="GROUP_6:func.6">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K6" title="GROUP_6:func.6">@</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K6" title="GROUP_6:func.6">@</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:30_1_1_1_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K6" title="GROUP_6:func.6">@</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K6" title="GROUP_6:func.6">@</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:30_1_1_1_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E4:30_1_1_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="group_2.html#T14" title="GROUP_2:th.14">GROUP_2:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K6" title="GROUP_6:func.6">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K6" title="GROUP_6:func.6">@</a> <font color="Maroon" title="c5">h</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:30_1_1_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h</font>
<b>by </b><i><a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</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>

<b>reconsider </b><font color="Maroon" title="c3">e</font> =  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">N</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font></span>)</span> <b>by </b><i><a class="ref" href="group_2.html#T165" title="GROUP_2:th.165">GROUP_2:165</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font></span>)</span> holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> ) )</span><br/>

<b>let </b><font color="Maroon" title="c4">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp;  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font></span>)</span> st <br/>( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> ) )</span><br/>

<b>consider </b><font color="Maroon" title="c5">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:30_1_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
( <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="txt" href="#E4:30_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T60" title="GROUP_4:th.60">GROUP_4:60</a></i>
<br/>.= 
<font color="Maroon" title="c4">h</font>
<b>by </b><i><a class="txt" href="#E4:30_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="group_2.html#T91" title="GROUP_2:th.91">GROUP_2:91</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp;  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font></span>)</span> st <br/>( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> ) )</span><br/>

<b>thus </b><font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> = 
<font color="Maroon" title="c2">N</font> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:30_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="group_4.html#T61" title="GROUP_4:th.61">GROUP_4:61</a></i>
<br/>.= 
<font color="Maroon" title="c4">h</font>
<b>by </b><i><a class="txt" href="#E4:30_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="group_2.html#T91" title="GROUP_2:th.91">GROUP_2:91</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font></span>)</span> st <br/>( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font></span>)</span> <b>by </b><i><a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">h</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="c3">e</font> &amp; <font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<b>thus </b><font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:30_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="ref" href="group_3.html#T10" title="GROUP_3:th.10">GROUP_3:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="ref" href="group_2.html#T129" title="GROUP_2:th.129">GROUP_2:129</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T132" title="GROUP_2:th.132">GROUP_2:132</a></i>
<br/>.= 
<font color="Maroon" title="c3">e</font>
<b>by </b><i><a class="ref" href="group_2.html#T91" title="GROUP_2:th.91">GROUP_2:91</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font></span><br/>

<b>thus </b><font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K6" title="GROUP_6:func.6">@</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K6" title="GROUP_6:func.6">@</a> <font color="Maroon" title="c4">h</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="txt" href="#E4:30_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="group_3.html#T10" title="GROUP_3:th.10">GROUP_3:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="ref" href="group_2.html#T128" title="GROUP_2:th.128">GROUP_2:128</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font></span>)</span></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="ref" href="group_3.html#T140" title="GROUP_3:th.140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="ref" href="group_2.html#T127" title="GROUP_2:th.127">GROUP_2:127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T60" title="GROUP_4:th.60">GROUP_4:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T91" title="GROUP_2:th.91">GROUP_2:91</a></i>
<br/>.= 
<font color="Maroon" title="c3">e</font>
<b>by </b><i><a class="ref" href="group_2.html#T43" title="GROUP_2:th.43">GROUP_2:43</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>
<b>hence </b><a NAME="E2:30_1_1"/>
( <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font> is <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
