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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><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="E1:19"/><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">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D</font>, <font color="Maroon">g</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D</font>, <font color="Maroon">h</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<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>set </b><font color="Maroon">F</font> = <font color="Maroon">o</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">o</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></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>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon">o</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">g</font>,<font color="Maroon">h</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">o</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">g</font>,<font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">o</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">o</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">g</font>,<font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
;<br/>


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