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

<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#V16" title="MONOID_0:attr.16">cancelable</a>  iff ( <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>  ) )</span><br/>

<span class="kw">thus </span><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>  ) )
  <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#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>  implies <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D19" target="_self" title="MONOID_0:def.19">MONOID_0:def 19</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#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>  )</span><br/>

<span class="kw">hence </span><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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T2" target="_self" title="MONOID_0:th.2">Th2</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><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> <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="E2:32"/>
( <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>  )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><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> <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#V16" title="MONOID_0:attr.16">cancelable</a> </span><br/>

<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T2" target="_self" title="MONOID_0:th.2">Th2</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D19" target="_self" title="MONOID_0:def.19">MONOID_0:def 19</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>
