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<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being 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> <br/> for <font color="Olive" title="b2">o</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">D</font>  st <font color="Olive" title="b2">o</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> holds <br/><font color="Olive" title="b2">o</font>,<font color="Olive" title="b1">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a></span><br/><b>let </b><font color="Maroon" title="c2">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">o</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> holds <br/><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a></span><br/><b>let </b><font color="Maroon" title="c3">o</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">o</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> implies <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )</span><br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c3">o</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a></span><br/>

<b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font>, <font color="Maroon" title="c6">h</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D14" title="BINOP_1:def.14">BINOP_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font></span><br/>

<b>set </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>
<b>by </b><i><a class="ref" href="monoid_1.html#D2" target="_self" title="MONOID_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c3">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>
<b>by </b><i><a class="ref" href="monoid_1.html#D2" target="_self" title="MONOID_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c3">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="monoid_1.html#T5" target="_self" title="MONOID_1:th.5">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="monoid_1.html#K3" title="MONOID_1:func.3">.:</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font></span>)</span>
<b>by </b><i><a class="ref" href="monoid_1.html#D2" target="_self" title="MONOID_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c2">D</font> <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font></span>)</span>
<b>by </b><i><a class="ref" href="monoid_1.html#D2" target="_self" title="MONOID_1:def.2">Def2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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