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

<b>let </b><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> ;<br/><b>let </b><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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">d1</font>, <font color="Maroon" title="c4">d2</font>, <font color="Maroon" title="c5">d3</font>, <font color="Maroon" title="c6">d4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/>







<b>thus </b><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">d2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d3</font>,<font color="Maroon" title="c6">d4</font></span>)</span> = 
<font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">d1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">d2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d3</font>,<font color="Maroon" title="c6">d4</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E2:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">d1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">d2</font>,<font color="Maroon" title="c5">d3</font></span>)</span>,<font color="Maroon" title="c6">d4</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E2:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">d1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d3</font>,<font color="Maroon" title="c4">d2</font></span>)</span>,<font color="Maroon" title="c6">d4</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">d1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">d2</font>,<font color="Maroon" title="c6">d4</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E2:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c5">d3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">d2</font>,<font color="Maroon" title="c6">d4</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E2:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>
;<br/>


</div>
