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

<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>assume </b><a NAME="E1:33"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V4">associative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="monoid_0.html#V13" target="_self">invertible</a> )
 ;<br/>

<a NAME="E2:33"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V2">unital</a>
 
<b>by </b><i><a class="ref" href="monoid_0.html#T17" target="_self">Th17</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:33"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T6" target="_self">Th6</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>thus </b><a NAME="E5:33"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E4:33"><i><font color="Green">E24</font></i></a></i>;<br/>

<a NAME="E6:33"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) is <a href="monoid_0.html#V5" target="_self">invertible</a>
 
<b>by </b><i><a class="txt" href="monoid_0.html#E1:33"><i><font color="Green">E23</font></i></a>, <a class="ref" href="monoid_0.html#D16" target="_self">Def16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:33"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="monoid_0.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E9:33"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="monoid_0.html#E8:33"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E4:33"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:33"><i><font color="Green">E23</font></i></a>, <a class="txt" href="monoid_0.html#E9:33"><i><font color="Green">E26</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>4</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E4:33"><i><font color="Green">E24</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E11:33"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E8:33"><i><font color="Green">E25</font></i></a></i>;<br/>


</div>
