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<span class="kw">let </span><font color="Maroon" title="c1">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font> holds <br/> ( <font color="Olive" title="b1">f</font> is  <a href="monoid_0.html#V8" title="MONOID_0:attr.8">cancelable</a>  iff ( <font color="Olive" title="b1">f</font> is  <a href="monoid_0.html#V6" title="MONOID_0:attr.6">left-cancelable</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="monoid_0.html#V7" title="MONOID_0:attr.7">right-cancelable</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V8" title="MONOID_0:attr.8">cancelable</a>  iff ( <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V6" title="MONOID_0:attr.6">left-cancelable</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V7" title="MONOID_0:attr.7">right-cancelable</a>  ) )</span><br/>



<span class="kw">thus </span><a NAME="E1:11"/>
( <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V8" title="MONOID_0:attr.8">cancelable</a>  implies ( <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V6" title="MONOID_0:attr.6">left-cancelable</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V7" title="MONOID_0:attr.7">right-cancelable</a>  ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V6" title="MONOID_0:attr.6">left-cancelable</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V7" title="MONOID_0:attr.7">right-cancelable</a>  implies <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V8" title="MONOID_0:attr.8">cancelable</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:11_1"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st ( <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> or <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b1">a</font> ) holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D8" target="_self" title="MONOID_0:def.8">MONOID_0:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V6" title="MONOID_0:attr.6">left-cancelable</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V7" title="MONOID_0:attr.7">right-cancelable</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E2:11_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D6" target="_self" title="MONOID_0:def.6">MONOID_0:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V7" title="MONOID_0:attr.7">right-cancelable</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:11_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b1">a</font> holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D7" target="_self" title="MONOID_0:def.7">MONOID_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E2:11"/>
( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b1">a</font> holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> ) )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D6" target="_self" title="MONOID_0:def.6">MONOID_0:def 6</a>,<a class="ref" href="monoid_0.html#D7" target="_self" title="MONOID_0:def.7">MONOID_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> is  <a href="monoid_0.html#V8" title="MONOID_0:attr.8">cancelable</a> </span><br/>

<span class="kw">hence </span><a NAME="E3:11"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st ( <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> or <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b1">a</font> ) holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D8" target="_self" title="MONOID_0:def.8">MONOID_0:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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