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

<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">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><b>that </b><br/><a NAME="E1:38"/><i><font color="Green">E14</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>and </b><br/><a NAME="E2:38"/><i><font color="Green">E15</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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>] <b>means </b>( ( 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">a<sub>1</sub></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> ) implies <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> );<br/>
<a NAME="E3:38"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Olive">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E21</font></i>: 
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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> 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>
 ;<br/>

<a NAME="E2:38_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>,<font color="Maroon">f</font> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="setwiseo.html#T12" target="_self">Th12</a>, <a class="ref" href="setwiseo.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="setwiseo.html#T14" target="_self">Th14</a>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:38"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">B1</font>, <font color="Olive">B2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B1</font> <a href="setwiseo.html#K5" target="_self">\/</a> <font color="Olive">B2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<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/>

<b>assume </b><a NAME="E1:38_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:38_2"/><i><font color="Green">E25</font></i>: 
( ( 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">B1</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> ) implies <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B1</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>and </b><br/><a NAME="E3:38_2"/><i><font color="Green">E51</font></i>: 
( ( for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) implies <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B2</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>and </b><br/><a NAME="E4:38_2"/><i><font color="Green">E57</font></i>: 
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">B1</font> <a href="setwiseo.html#K5" target="_self">\/</a> <font color="Maroon">B2</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>
 ;<br/>

<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><b>now </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="E1:38_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 ;<br/><a NAME="E2:38_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="setwiseo.html#K5" target="_self">\/</a> <font color="Maroon">B2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:38_2_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="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_2"><b>now </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="E1:38_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 ;<br/><a NAME="E2:38_2_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="setwiseo.html#K5" target="_self">\/</a> <font color="Maroon">B2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:38_2_2"/>
<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="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div>
<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> <font color="Maroon">a</font>,<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#T16" target="_self">Th16</a>, <a class="ref" href="setwiseo.html#T18" target="_self">Th18</a>, <a class="txt" href="setwiseo.html#E4:38"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="setwiseo.html#T12" target="_self">Th12</a>, <a class="ref" href="binop_1.html#D4">BINOP_1:def 4</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E5:38"/>
for <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S3" target="_self">SETWISEO:sch 3</a>(<a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, <a class="ref" href="setwiseo.html#T14" target="_self">Th14</a>);<br/></i>
<b>hence </b><a NAME="E6:38"/>
( ( 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> ) implies <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/>;<br/>


</div>
