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<div class="add">

<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">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">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">B</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/><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><a NAME="E1:39"/><i><font color="Green">E14</font></i>: 
( <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> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V3">idempotent</a> )
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E2:39"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:39"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/>
<a NAME="E4:39"/>
for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:39_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:39"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T12" target="_self">Th12</a>, <a class="ref" href="setwiseo.html#T14" target="_self">Th14</a>, </i>;<br/>


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