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

<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span> ;<br/>
<a NAME="E1:88_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font>
 
;<br/>
<a NAME="E2:88_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1_1"><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="E1:88_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span> 
 ;<br/>

<a NAME="E2:88_1_1_1"/><b>then </b>
 ex <font color="Olive">c</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:88_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>set </b><font color="Maroon">H</font> = <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> as   <a href="armstrng.html#NM5">Full-family</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E5:88_1_1"/>
( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> &amp; ( for <font color="Olive">G</font> being  <a href="armstrng.html#NM2">Dependency-set</a> of <font color="Maroon">X</font>  st <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> is <a href="armstrng.html#V6" target="_self">full_family</a> holds <br/><font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> ) )
 <b>by </b><i>, </i>;<br/>


</div>
