<?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>, <font color="Maroon">G</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">Y</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:42"/><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="E2:42"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">G</font> <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">F</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>assume </b><a NAME="E3:42"/><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> 
 ;<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>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><font color="Maroon">G</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">$$</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>;<br/>
<a NAME="E4:42"/><i><font color="Green">E21</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="42_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>thus </b><font color="Maroon">G</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">$$</a> <span class="p2"><a href="setwiseo.html#K2" target="_self">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2" target="_self">.}</a></span>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, <a class="ref" href="setwiseo.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T60">FUNCOP_1:60</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</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>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, <a class="ref" href="setwiseo.html#T11" target="_self">Th11</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:42"/><i><font color="Green">E22</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="42_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="E1:42_2"/><i><font color="Green">E23</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="E2:42_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">G</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">$$</a> <font color="Maroon">B1</font>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B1</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
 <b>and </b><br/><a NAME="E3:42_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">G</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">$$</a> <font color="Maroon">B2</font>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B2</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
 ;<br/>

<b>thus </b><font color="Maroon">G</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">$$</a> <span class="p2">(<span class="default"><font color="Maroon">B1</font> <a href="setwiseo.html#K5" target="_self">\/</a> <font color="Maroon">B2</font></span>)</span>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B1</font>,<font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B2</font>,<font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, <a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="txt" href="setwiseo.html#E1:42_2"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B1</font>,<span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B2</font>,<span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T14" target="_self">Th14</a>, <a class="ref" href="setwiseo.html#T18" target="_self">Th18</a>, <a class="ref" href="setwiseo.html#T19" target="_self">Th19</a>, <a class="ref" href="binop_1.html#T23">BINOP_1:23</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</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>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, <a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="txt" href="setwiseo.html#E1:42_2"><i><font color="Green">E23</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:42"/>
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#T16" target="_self">Th16</a>);<br/></i>
<b>hence </b><a NAME="E7:42"/>
<font color="Maroon">G</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">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
