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<b>let </b><font color="Maroon" title="c1">G</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> ;<br/>

<b>thus </b><a NAME="E1:32"/>
( <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a> implies ( <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V14" title="MONOID_0:attr.14">left-cancelable</a> &amp; <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V15" title="MONOID_0:attr.15">right-cancelable</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V8" title="MONOID_0:attr.8">cancelable</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D19" target="_self" title="MONOID_0:def.19">MONOID_0:def 19</a><br/>

<b>hence </b><a NAME="E2:32_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V6" title="MONOID_0:attr.6">left-cancelable</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V7" title="MONOID_0:attr.7">right-cancelable</a> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T2" target="_self" title="MONOID_0:th.2">Th2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D17" target="_self" title="MONOID_0:def.17">MONOID_0:def 17</a>,<a class="ref" href="monoid_0.html#D18" target="_self" title="MONOID_0:def.18">MONOID_0:def 18</a><br/>


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

<b>assume </b><a NAME="E2:32"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V6" title="MONOID_0:attr.6">left-cancelable</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V7" title="MONOID_0:attr.7">right-cancelable</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D17" target="_self" title="MONOID_0:def.17">MONOID_0:def 17</a>,<a class="ref" href="monoid_0.html#D18" target="_self" title="MONOID_0:def.18">MONOID_0:def 18</a><br/>

<b>hence </b><a NAME="E3:32"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V8" title="MONOID_0:attr.8">cancelable</a>
 <b>by </b><i><a class="ref" href="monoid_0.html#T2" target="_self" title="MONOID_0:th.2">Th2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D19" target="_self" title="MONOID_0:def.19">MONOID_0:def 19</a><br/>


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