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<b>let </b><font color="Maroon">G</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">H</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">G</font>;<br/>



<b>assume </b><a NAME="E1:87"/><i><font color="Green">E28</font></i>: 
 <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>)
 ;<br/>

<b>assume </b><a NAME="E2:87"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  st <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D9" target="_self">MONOID_0:def 9</a>,<a class="ref" href="monoid_0.html#D20" target="_self">MONOID_0:def 20</a><br/>

<a NAME="E3:87"/><b>then </b>
<font color="Maroon">G</font> is <a href="group_1.html#V2">unital</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:87"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">H</font> is <a href="group_1.html#V2">unital</a> &amp;  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">H</font>) )
 
<b>by </b><i><a class="ref" href="monoid_0.html#D10" target="_self">Def10</a>, <a class="txt" href="monoid_0.html#E4:87"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:87"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">H</font>) <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D9" target="_self">MONOID_0:def 9</a>,<a class="ref" href="monoid_0.html#D20" target="_self">MONOID_0:def 20</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>;<br/>

<a NAME="E6:87"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>)
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">H</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>

<br/>.= 
<font color="Maroon">a'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>

;<br/>
<b>hence </b><a NAME="E9:87"/>
( the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H</font> implies ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H</font> ) )
 <b>by </b><i>, </i>;<br/>


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