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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="monoid_0.html#M2" target="_self">SubStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:84"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V4">associative</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 <b>by </b><i><a class="ref" href="monoid_0.html#T25" target="_self">Th25</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="monoid_0.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="monoid_0.html#T27" target="_self">Th27</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:84"><i><font color="Green">E44</font></i></a>, <a class="txt" href="monoid_0.html#E28"><i><font color="Green">Lemma14</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="monoid_0.html#T27" target="_self">Th27</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="monoid_0.html#T27" target="_self">Th27</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:84"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="group_1.html#V4">associative</a>
 <b>by </b><i><a class="txt" href="monoid_0.html#E28"><i><font color="Green">Lemma14</font></i></a></i>;<br/>


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