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

<b>let </b><font color="Maroon">D</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E1:39"/>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:39"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:39"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/>( ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> ) iff for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>thus </b><a NAME="E2:39_1"/>
( ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> ) implies for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:39_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> ;<br/>
<a NAME="E4:39_1_1"/>
<font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:39_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:39_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font>
 
;<br/>
<b>hence </b><a NAME="E7:39_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</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>assume </b><a NAME="E3:39_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 ;<br/>

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

<b>assume </b><a NAME="E4:39_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> ;<br/>
<a NAME="E6:39_1"/>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E7:39_1"/><b>then </b>
<font color="Maroon">X</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:39_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:39_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">X</font>
 
;<br/>
<b>hence </b><a NAME="E10:39_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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>
<a NAME="E4:39"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" target="_self">union</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6" target="_self">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" target="_self">union</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<a NAME="E2:39_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">D</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5" target="_self">union</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:39_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:39_2"/><b>then </b>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:39_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6" target="_self">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:39"/>
 <a href="setfam_1.html#K6" target="_self">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" target="_self">union</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6" target="_self">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<a NAME="E2:39_3"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
;<br/>
<a NAME="E3:39_3"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:39_3"/><b>then </b>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> holds <br/> not <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:39_3"/><b>then </b><i><font color="Green">E36</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5" target="_self">union</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:39_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:39_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" target="_self">union</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:39"/>
<span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" target="_self">union</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6" target="_self">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7" target="_self">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
