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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="armstrng.html#NM5" title="ARMSTRNG:NM.5">Full-family</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>set </b><font color="Maroon" title="c3">G</font> =  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>;<br/>
<b>take </b>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
;<br/>

<a NAME="E1:95"/><i><font color="Green" title="E55">A1</font></i>: 
(  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a> &amp;  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">(B2)</a> )
 
<b>by </b><i><a class="ref" href="armstrng.html#T34" target="_self" title="ARMSTRNG:th.34">Th34</a></i>;<br/>
<b>thus </b><a NAME="E2:95"/>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:95_1"/>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D26" target="_self" title="ARMSTRNG:def.26">ARMSTRNG:def 26</a><br/>

<b>set </b><font color="Maroon" title="c4">H</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>set </b><font color="Maroon" title="c6">sx</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>assume </b><a NAME="E1:95_1_1_1"/><i><font color="Green" title="E56">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 ;<br/><a NAME="E2:95_1_1_1"/><b>then </b><i><font color="Green" title="E57">A3</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">sx</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="armstrng.html#E1:95_1_1_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E4:95_1_1_1"/><i><font color="Green" title="E58">A4</font></i>: 
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">sx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">sx</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/><a NAME="E5:95_1_1_1"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">sx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E2:95_1_1_1"><i><font color="Green" title="E57">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E6:95_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E4:95_1_1_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:95_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c6">S</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:95_1_1"/><i><font color="Green" title="E56">A5</font></i>: 
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <b>and </b><br/><a NAME="E5:95_1_1"/><i><font color="Green" title="E57">A6</font></i>: 
<font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 ;<br/><a NAME="E6:95_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:95"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="armstrng.html#D4" target="_self" title="ARMSTRNG:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E7:95_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:95"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="armstrng.html#E4:95_1_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="txt" href="armstrng.html#E5:95_1_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="ref" href="armstrng.html#T1" target="_self" title="ARMSTRNG:th.1">Th1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:95_1"/>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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

<b>thus </b><a NAME="E3:95"/>
<font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">(<span class="default"><a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:95"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="armstrng.html#T37" target="_self" title="ARMSTRNG:th.37">Th37</a></i>;<br/>


</div>
