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<b>let </b><font color="Maroon" title="c1">G</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ;<br/><b>let </b><font color="Maroon" title="c2">H</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>;<br/><a NAME="E1:81_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">H</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>)
 <b>by </b><i><a class="ref" href="monoid_0.html#T25" target="_self" title="MONOID_0:th.25">Th25</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">b'</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">c'</font> = <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c9">ab</font> = <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c10">bc</font> = <font color="Maroon" title="c4">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c</font> = 
<font color="Maroon" title="c9">ab</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">c'</font>
<b>by </b><i><a class="ref" href="monoid_0.html#T27" target="_self" title="MONOID_0:th.27">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b'</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">c'</font>
<b>by </b><i><a class="ref" href="monoid_0.html#T27" target="_self" title="MONOID_0:th.27">Th27</a></i>
<br/>.= 
<font color="Maroon" title="c6">a'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:81"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="monoid_0.html#E28"><i><font color="Green" title="E14">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">a'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">bc</font>
<b>by </b><i><a class="ref" href="monoid_0.html#T27" target="_self" title="MONOID_0:th.27">Th27</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c</font></span>)</span>
<b>by </b><i><a class="ref" href="monoid_0.html#T27" target="_self" title="MONOID_0:th.27">Th27</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:81"/>
<font color="Maroon" title="c2">H</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>
 <b>by </b><i><a class="txt" href="monoid_0.html#E28"><i><font color="Green" title="E14">Lm2</font></i></a></i>;<br/>


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