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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>thus </b><a NAME="E1:31"/>
( <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a> implies ( <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V11" title="MONOID_0:attr.11">left-invertible</a> &amp; <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V12" title="MONOID_0:attr.12">right-invertible</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V5" title="MONOID_0:attr.5">invertible</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a><br/>

<b>hence </b><a NAME="E2:31_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V3" title="MONOID_0:attr.3">left-invertible</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V4" title="MONOID_0:attr.4">right-invertible</a> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T1" target="_self" title="MONOID_0:th.1">Th1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D14" target="_self" title="MONOID_0:def.14">MONOID_0:def 14</a>,<a class="ref" href="monoid_0.html#D15" target="_self" title="MONOID_0:def.15">MONOID_0:def 15</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:31"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V3" title="MONOID_0:attr.3">left-invertible</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V4" title="MONOID_0:attr.4">right-invertible</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D14" target="_self" title="MONOID_0:def.14">MONOID_0:def 14</a>,<a class="ref" href="monoid_0.html#D15" target="_self" title="MONOID_0:def.15">MONOID_0:def 15</a><br/>

<b>hence </b><a NAME="E3:31"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V5" title="MONOID_0:attr.5">invertible</a>
 <b>by </b><i><a class="ref" href="monoid_0.html#T1" target="_self" title="MONOID_0:th.1">Th1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a><br/>


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