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<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">o</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E1:79"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">o</font> is <a href="binop_1.html#V2">associative</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>, <font color="Maroon">y</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>, <font color="Maroon">z</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D14">BINOP_1:def 14</a><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>,<font color="Maroon">z</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
;<br/>


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