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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  iff ( <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>  ) )</span><br/>

<span class="kw">thus </span><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>  ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  implies <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  )</span><br/>

<span class="kw">hence </span><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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T1" target="_self" title="MONOID_0:th.1">Th1</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><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> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><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>  )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><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> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a> </span><br/>

<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T1" target="_self" title="MONOID_0:th.1">Th1</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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