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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="armstrng.html#NM2">Dependency-set</a> of <font color="Maroon">X</font>;<br/>



<b>set </b><font color="Maroon">B</font> =  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</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> <span class="p1">(<span class="default"><a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font></span>)</span>;<br/>
<b>thus </b><a NAME="E1:87"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_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="E1:87_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">b</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:87_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:87_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:87_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E3:87_1_1"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 ;<br/><b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E5:87_1_1"/>
 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="Olive">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:87_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="armstrng.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:87_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">G</font> be   <a href="armstrng.html#NM2">Dependency-set</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:87"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E3:87"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G</font> is <a href="armstrng.html#V6" target="_self">full_family</a>
 ;<br/>

<b>let </b><font color="Maroon">z</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="E4:87"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">b</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:87"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E7:87"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font>
 ;<br/>
<a NAME="E8:87"/>
(  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font> is <a href="armstrng.html#V1" target="_self">(B1)</a> &amp;  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font> is <a href="finsub_1.html#V2">(B2)</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:87"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <span class="p2">(<span class="default"><a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">B'</font> =  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">G</font>;<br/>
<a NAME="E10:87"/><i><font color="Green">E50</font></i>: 
(  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">G</font> is <a href="armstrng.html#V1" target="_self">(B1)</a> &amp;  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">G</font> is <a href="finsub_1.html#V2">(B2)</a> )
 
<b>by </b><i>, <a class="txt" href="armstrng.html#E7:87"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">GG</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">e</font>,<font color="Olive">f</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">e</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">f</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">G</font> &amp; <font color="Olive">e</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span> ;<br/>
<a NAME="E11:87"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">e</font>,<font color="Olive">f</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">e</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">f</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">G</font> &amp; <font color="Olive">e</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">G</font></span>)</span>
 
;<br/>
<a NAME="E12:87"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">e</font>,<font color="Olive">f</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">e</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">f</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">G</font> &amp; <font color="Olive">e</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:87"/>
 <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <span class="p2">(<span class="default"><a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</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><b>that </b><br/><a NAME="E1:87_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E2:87_2"/><i><font color="Green">E79</font></i>: 
not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <span class="p2">(<span class="default"><a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:87_2"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E6:87_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E7:87_2"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:87_2"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">e</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">f</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E10:87_2"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">f</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E11:87_2"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">G</font> &amp; <font color="Maroon">e</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font>
 <b>by </b><i/>;<br/>
<a NAME="E12:87_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E13:87_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="armstrng.html#D1" target="_self">Def1</a>, <a class="ref" href="armstrng.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:87"/><b>then </b>
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font>
 
<b>by </b><i><a class="ref" href="armstrng.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>hence </b><a NAME="E15:87"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/>


</div>
