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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">S</font> is   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">X</font>
 ;<br/><a NAME="E2:16_1"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/>

<a NAME="E2:16_1_1"/><b>then </b>
( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:16_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="finsub_1.html#D1">FINSUB_1:def 1</a></i>;<br/>
<a NAME="E4:16_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E5:16_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then </b><font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> = 
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T48">XBOOLE_1:48</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>
;<br/>
<b>hence </b><a NAME="E7:16_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_1"/>
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/>( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:16"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/>( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 ;<br/>

<a NAME="E3:16"/><b>then </b><i><font color="Green">E27</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
;<br/>
<a NAME="E4:16"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/>

<a NAME="E2:16_2"/><b>then </b>
( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:16_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:16_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>then </b><font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> = 
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T48">XBOOLE_1:48</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T53">XBOOLE_1:53</a></i>
;<br/>
<b>hence </b><a NAME="E6:16_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:16"/><b>then </b>
( <font color="Maroon">S</font> is <a href="finsub_1.html#V1">cup-closed</a> &amp; <font color="Maroon">S</font> is <a href="prob_1.html#V1">compl-closed</a> )
 
<b>by </b><i>, , <a class="ref" href="finsub_1.html#D1">FINSUB_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:16"/>
<font color="Maroon">S</font> is   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div>
