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<b>let </b><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">X</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="c3">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">X</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c2">X</font>;<br/>









<b>assume </b><a NAME="E1:80"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> = 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T48" target="_self" title="FUNCOP_1:th.48">Th48</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span>
<b>by </b><i><a class="txt" href="funcop_1.html#E1:80"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:80"/>
<font color="Maroon" title="c3">F</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">F</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T49" target="_self" title="FUNCOP_1:th.49">Th49</a></i>;<br/>


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