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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:50"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></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#T64">RELAT_1:64</a></i>;<br/>
<a NAME="E3:50"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:50_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_3.html#T55" target="_self">Th55</a></i>;<br/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="txt" href="prob_3.html#E2:50_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="prob_3.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>

;<br/>
<b>hence </b><a NAME="E4:50_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:50"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K8" target="_self">Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  iff <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="prob_3.html#K8" target="_self">Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_3.html#K8" target="_self">Union</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> iff ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<a NAME="E1:50_2"/><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="prob_3.html#T55" target="_self">Th55</a></i>;<br/>
<b>hence </b><a NAME="E2:50_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_3.html#K8" target="_self">Union</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> implies ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="ref" href="prob_3.html#T54" target="_self">Th54</a></i>;<br/>

<b>thus </b><a NAME="E3:50_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) implies ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_3.html#K8" target="_self">Union</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="prob_3.html#E1:50_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="prob_3.html#T54" target="_self">Th54</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:50"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) iff for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:50_3_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E2:50_3_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E3:50_3_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="prob_3.html#E2:50_3_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="prob_3.html#E1:50_3_1"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E4:50_3_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:50_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E2:50_3_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="prob_3.html#E3:50"><i><font color="Green">E46</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:50_3"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:50_3_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:50_3_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:50_3_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="prob_3.html#E2:50_3"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E3:50_3_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:50_3_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="prob_3.html#E3:50"><i><font color="Green">E46</font></i></a></i>;<br/><b>hence </b><a NAME="E4:50_3_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:50_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="prob_3.html#E2:50"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>thus </b><a NAME="E6:50_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K9" target="_self">Complement</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="prob_3.html#E5:50_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="prob_3.html#E3:50_3"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:50"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_3.html#K10" target="_self">Intersection</a> <font color="Maroon">c<sub>3</sub></font> iff for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="prob_3.html#E1:50"><i><font color="Green">E44</font></i></a>, <a class="txt" href="prob_3.html#E4:50"><i><font color="Green">E47</font></i></a>, <a class="txt" href="prob_3.html#E5:50"><i><font color="Green">E48</font></i></a>, <a class="ref" href="prob_3.html#D9" target="_self">Def9</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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