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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>



<b>thus </b><a NAME="E1:11"/>
( <font color="Maroon">f</font> is <a href="monoid_0.html#V8" target="_self">cancelable</a> implies ( <font color="Maroon">f</font> is <a href="monoid_0.html#V6" target="_self">left-cancelable</a> &amp; <font color="Maroon">f</font> is <a href="monoid_0.html#V7" target="_self">right-cancelable</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st ( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">c</font> or <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">c</font>,<font color="Olive">a</font> ) holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D8" target="_self">MONOID_0:def 8</a><br/>

<b>hence </b><a NAME="E2:11_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">c</font> holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D6" target="_self">MONOID_0:def 6</a><br/>

<b>thus </b><a NAME="E3:11_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">c</font>,<font color="Olive">a</font> holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D7" target="_self">MONOID_0:def 7</a><br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:11"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">c</font> holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 <b>and </b><br/><a NAME="E3:11"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">c</font>,<font color="Olive">a</font> holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D6" target="_self">MONOID_0:def 6</a>,<a class="ref" href="monoid_0.html#D7" target="_self">MONOID_0:def 7</a><br/>

<b>thus </b><a NAME="E4:11"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st ( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">c</font> or <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">c</font>,<font color="Olive">a</font> ) holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D8" target="_self">MONOID_0:def 8</a><br/>


</div>
