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



<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:111_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V3" target="_self">(F1)</a>
 <b>and </b><br/><a NAME="E2:111_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> is <a href="relat_2.html#V8">(F2)</a>
 <b>and </b><br/><a NAME="E3:111_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V4" target="_self">(F3)</a>
 <b>and </b><br/><a NAME="E4:111_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V5" target="_self">(F4)</a>
 ;<br/><b>thus </b><a NAME="E5:111_1"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V3" target="_self">(F1)</a>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:111_1"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V13" target="_self">(DC5)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1"><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">C</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">D</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#D30" target="_self">ARMSTRNG:def 30</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:111_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> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:111_1_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">D</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E3:111_1_1"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">C</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">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:111_1_1"/><b>then </b>
<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">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="armstrng.html#D2" target="_self">Def2</a>, <a class="ref" href="armstrng.html#D3" target="_self">Def3</a>, </i>;<br/>
<b>hence </b><a NAME="E5:111_1_1"/>
<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">C</font></span>)</span>,<font color="Maroon">D</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D1" target="_self">Def1</a>, , </i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E7:111_1"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V14" target="_self">(DC6)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_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>, <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D31" target="_self">ARMSTRNG:def 31</a><br/>

<b>assume </b><a NAME="E1:111_1_2"/><i><font color="Green">E43</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">F</font>
 ;<br/>

<a NAME="E2:111_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V7" target="_self">(DC3)</a>
 
<b>by </b><i>, , <a class="txt" href="armstrng.html#E1:111_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E3:111_1_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:111_1_2"/><b>then </b>
<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">C</font></span>)</span>,<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">F</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:111_1_2"/>
<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">C</font></span>)</span>,<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">F</font>
 <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>

<b>assume </b><b>that </b><br/><a NAME="E2:111"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V3" target="_self">(F1)</a>
 <b>and </b><br/><a NAME="E3:111"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V13" target="_self">(DC5)</a>
 <b>and </b><br/><a NAME="E4:111"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V14" target="_self">(DC6)</a>
 ;<br/>

<b>thus </b><a NAME="E5:111"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V3" target="_self">(F1)</a>
 <b>by </b><i/>;<br/>

<div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><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="E1:111_2"/><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">B</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="E2:111_2"/><i><font color="Green">E79</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">F</font>
 ;<br/><a NAME="E3:111_2"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font></span>)</span>,<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">F</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:111_2"/><b>then </b>
<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>,<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">F</font>
 <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/><b>hence </b><a NAME="E5:111_2"/>
<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">F</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:111"/>
<font color="Maroon">F</font> is <a href="relat_2.html#V8">(F2)</a>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E8:111"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V4" target="_self">(F3)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_3"><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="E1:111_3"/><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">F</font>
 <b>and </b><br/><a NAME="E2:111_3"/><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="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="E3:111_3"/><i><font color="Green">E82</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/>
<a NAME="E4:111_3"/><b>then </b>
<font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E5:111_3"/><b>then </b><i><font color="Green">E83</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">F</font>
 
<b>by </b><i>, <a class="ref" href="armstrng.html#D3" target="_self">Def3</a>, </i>;<br/>
<a NAME="E6:111_3"/><i><font color="Green">E84</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">B'</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">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:111_3"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E8:111_3"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">B</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">F</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E9:111_3"/>
<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">F</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D2" target="_self">Def2</a>, </i>;<br/>


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

<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="E9:111"/><i><font color="Green">E85</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">F</font>
 <b>and </b><br/><a NAME="E10:111"/><i><font color="Green">E109</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">F</font>
 ;<br/>

<a NAME="E11:111"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><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 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">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:111"/><b>then </b>
<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">B'</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">F</font>
 
<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/>
<b>hence </b><a NAME="E13:111"/>
<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">F</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D1" target="_self">Def1</a>, , </i>;<br/>


</div>
