<?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">G</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<b>set </b><font color="Maroon">F</font> = <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">ssF</font> =  <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> <font color="Maroon">G</font></span>)</span>;<br/>
<a NAME="E1:94"/>
<font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">G</font> is <a href="armstrng.html#V6" target="_self">full_family</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:94"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <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> <font color="Maroon">G</font></span>)</span> is <a href="armstrng.html#V1" target="_self">(B1)</a> &amp;  <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> <font color="Maroon">G</font></span>)</span> is <a href="finsub_1.html#V2">(B2)</a> )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:94"/>
<font color="Maroon">G</font> <a href="armstrng.html#R5" target="_self">is_generator-set_of</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> <font color="Maroon">G</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:94_1"/><i><font color="Green">E34</font></i>: 
<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> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D26" target="_self">ARMSTRNG:def 26</a><br/>

<b>set </b><font color="Maroon">H</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:94_1_1_1"/>
<font color="Maroon">x</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> <font color="Maroon">G</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">b'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">a'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:94_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 <b>and </b><br/><a NAME="E4:94_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a'</font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">b'</font>,<font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E10:94_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E5:94_1_1_1"/><i><font color="Green">E39</font></i>: 
<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> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="armstrng.html#E3:94_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E6:94_1_1_1"/>
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<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="E8:94_1_1_1"/><i><font color="Green">E40</font></i>: 
<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> <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="E9:94_1_1_1"/><i><font color="Green">E43</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> <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>
 ;<br/><a NAME="E10:94_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>set </b><font color="Maroon">C</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> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> ) </span>}</span> ;<br/><a NAME="E11:94_1_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> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</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="94_1_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="E1:94_1_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> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:94_1_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; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> )
 
;<br/>
<b>hence </b><a NAME="E3:94_1_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">C</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> : ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</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">ic</font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">C</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:94_1_1_1_2"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:94_1_1_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> )
 ;<br/><b>thus </b><a NAME="E4:94_1_1_1_2"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">z1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E14:94_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#T4">MSSUBFAM:4</a></i>;<br/><a NAME="E15:94_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</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="E1:94_1_1_1_3"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:94_1_1_1_3"/><b>then </b>
 ex <font color="Olive">cc</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">cc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Olive">cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">cc</font> )
 
;<br/>
<b>hence </b><a NAME="E3:94_1_1_1_3"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E16:94_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:94_1_1_1_4_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">C</font> or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:94_1_1_1_4_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:94_1_1_1_4_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> </span>}</span> 
 <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:94_1_1_1_4_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">ic</font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">C</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_4_1_2_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><a NAME="E1:94_1_1_1_4_1_2_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E2:94_1_1_1_4_1_2_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><b>hence </b><a NAME="E3:94_1_1_1_4_1_2_1"/>
<font color="Maroon">ic</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#T2">MSSUBFAM:2</a></i>;<br/></div><b>end;</b></div><a NAME="E4:94_1_1_1_4_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">ic</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E5:94_1_1_1_4_1_2"/>
<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> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">ic</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:94_1_1_1_4_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="armstrng.html#D1" target="_self">Def1</a>, <a class="ref" href="armstrng.html#D3" target="_self">Def3</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:94_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">S</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:94_1_1"/><i><font color="Green">E79</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:94_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E6:94_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">S</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> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:94_1_1"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T24">FINSET_1:24</a></i>;<br/><a NAME="E8:94_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">S</font> is  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E9:94_1_1"/>
<font color="Maroon">X</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> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E10:94_1_1"/>
<font color="Maroon">x</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> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:94_1"/>
 <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> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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


</div>
