<?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">B</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<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">F</font> = <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1"/>
(  not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">(C1)</a> or <font color="Maroon">B</font> is <a href="xboole_0.html#V1">(C1)</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><b>suppose </b></a><a NAME="E1:82_1_1"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">(C1)</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_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/><b>thus </b><a NAME="E1:82_1_1_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> <font color="Maroon">B</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> )
 ;<br/><b>assume </b><a NAME="E2:82_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:82_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E5:82_1_1_1"/>
for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">g</font> holds <br/><font color="Maroon">x2</font> <a href="tarski.html#R1">c=</a> <font color="Olive">g</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:82_1_1_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> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:82_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E4:82_1_1"/><b>then </b>
( <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> is <a href="armstrng.html#V3" target="_self">(F1)</a> &amp; <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> is <a href="relat_2.html#V8">(F2)</a> &amp; <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> is <a href="armstrng.html#V4" target="_self">(F3)</a> &amp; <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> is <a href="armstrng.html#V5" target="_self">(F4)</a> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:82_1_1"/>
<font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> is <a href="armstrng.html#V6" target="_self">full_family</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><b>suppose </b></a><a NAME="E1:82_1_2"/>
<font color="Maroon">B</font> is <a href="xboole_0.html#V1">(C1)</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  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><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:82_1_2_1"/><i><font color="Green">E38</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">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:82_1_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 ;<br/><b>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="E4:82_1_2_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">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">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E5:82_1_2_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">B</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/>;<br/><a NAME="E6:82_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</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>hence </b><a NAME="E7:82_1_2_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>thus </b><a NAME="E4:82_1_2"/>
<font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> is <a href="armstrng.html#V3" target="_self">(F1)</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D15" target="_self">ARMSTRNG:def 15</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D12" target="_self">ARMSTRNG:def 12</a><br/>

<a NAME="E1:82_1_2_2"/>
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">B</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font>
 
;<br/>
<b>hence </b><a NAME="E2:82_1_2_2"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</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">B</font>
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">b</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">c</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:82_1_2_3"/><i><font color="Green">E44</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> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:82_1_2_3"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</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">B</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_3_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/><b>assume </b><a NAME="E1:82_1_2_3_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 ;<br/><a NAME="E2:82_1_2_3_1"/><b>then </b>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:82_1_2_3_1"/>
<font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:82_1_2_3"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</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">B</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:82_1_2"/>
<font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> is <a href="relat_2.html#V8">(F2)</a>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E7:82_1_2"/>
<font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> is <a href="armstrng.html#V4" target="_self">(F3)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">b</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">a'</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">b'</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D13" target="_self">ARMSTRNG:def 13</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:82_1_2_4"/><i><font color="Green">E50</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> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:82_1_2_4"/><i><font color="Green">E74</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="armstrng.html#R3" target="_self">&gt;=</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>
 ;<br/>

<a NAME="E3:82_1_2_4"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_4_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="E1:82_1_2_4_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E2:82_1_2_4_1"/><b>then </b>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:82_1_2_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:82_1_2_4_1"/><b>then </b>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:82_1_2_4_1"/>
<font color="Maroon">b'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:82_1_2_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:82_1_2_4"/>
<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">B</font>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E8:82_1_2"/>
<font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> is <a href="armstrng.html#V5" target="_self">(F4)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">b</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">a'</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">b'</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D14" target="_self">ARMSTRNG:def 14</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:82_1_2_5"/><i><font color="Green">E80</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> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:82_1_2_5"/><i><font color="Green">E81</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> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_5_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="E1:82_1_2_5_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">a'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E2:82_1_2_5_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T11">XBOOLE_1:11</a></i>;<br/><a NAME="E3:82_1_2_5_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E4:82_1_2_5_1"/>
<font color="Maroon">b</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">b'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:82_1_2_5"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">a'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">b'</font></span>)</span></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">B</font>
 ;<br/>


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

</div>
