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<b>let </b><font color="Maroon">Y</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">X</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">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">F</font> is <a href="binop_1.html#V3">idempotent</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> )
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 ;<br/>

<b>let </b><font color="Maroon">B1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>, <font color="Maroon">B2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E2:48_1"/><i><font color="Green">E21</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1"/>
( <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><i><a class="ref" href="setwiseo.html#T18" target="_self">Th18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="setwiseo.html#K5" target="_self">\/</a> <font color="Maroon">B2</font></span>)</span>,<font color="Maroon">f</font> = 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B1</font>,<font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="txt" href="setwiseo.html#E1:48_1"><i><font color="Green">E16</font></i></a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B1</font>,<font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B2</font>,<font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T16" target="_self">Th16</a>, <a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="setwiseo.html#T19" target="_self">Th19</a>, , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="setwiseo.html#K5" target="_self">\/</a> <font color="Maroon">B2</font></span>)</span>,<font color="Maroon">f</font> = 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B2</font>,<font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="txt" href="setwiseo.html#E1:48_1"><i><font color="Green">E16</font></i></a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B1</font>,<font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B2</font>,<font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T16" target="_self">Th16</a>, <a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="setwiseo.html#T19" target="_self">Th19</a>, , </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="setwiseo.html#K5" target="_self">\/</a> <font color="Maroon">B2</font></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B1</font>,<font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B2</font>,<font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="setwiseo.html#T16" target="_self">Th16</a>, </i>;<br/>


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