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

<b>let </b><font color="Maroon">D</font> be    <a href="armstrng.html#L1" target="_self">DB-Rel</a> ;<br/>

<b>set </b><font color="Maroon">S</font> =  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">T</font> = the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">R</font> = the <a href="armstrng.html#U3" target="_self">Relationship</a> of <font color="Maroon">D</font>;<br/>
<a NAME="E1:65"/><i><font color="Green">E33</font></i>: 
 <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font> is <a href="armstrng.html#V7" target="_self">(DC3)</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_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 the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D16" target="_self">ARMSTRNG:def 16</a><br/>

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:65_1"/>
<font color="Maroon">A</font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">B</font>,<font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="armstrng.html#U3" target="_self">Relationship</a> of <font color="Maroon">D</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="armstrng.html#U3" target="_self">Relationship</a> of <font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D8" target="_self">ARMSTRNG:def 8</a><br/>

<b>assume </b><a NAME="E1:65_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font>
 ;<br/>

<b>thus </b><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font>
<b>by </b><i>, , <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:65_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>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font>
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_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 the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font>, <font color="Maroon">C</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:65_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="hidden.html#R2">in</a>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font> &amp; <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>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font> )
 ;<br/><a NAME="E2:65_2"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">A</font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">B</font>,<font color="Maroon">D</font> &amp; <font color="Maroon">B</font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">C</font>,<font color="Maroon">D</font> )
 <b>by </b><i/>;<br/><a NAME="E3:65_2"/>
<font color="Maroon">A</font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">C</font>,<font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="armstrng.html#U3" target="_self">Relationship</a> of <font color="Maroon">D</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="armstrng.html#U3" target="_self">Relationship</a> of <font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D8" target="_self">ARMSTRNG:def 8</a><br/>

<b>assume </b><a NAME="E1:65_2_1"/>
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:65_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:65_2_1"/>
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">C</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:65_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>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E3:65"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font> is <a href="relat_2.html#V8">(F2)</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:65"/>
 <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font> is <a href="armstrng.html#V3" target="_self">(F1)</a>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D15" target="_self">ARMSTRNG:def 15</a><br/>

<b>thus </b><a NAME="E5:65"/>
 <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font> is <a href="relat_2.html#V8">(F2)</a>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E6:65"/>
 <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font> is <a href="armstrng.html#V4" target="_self">(F3)</a>
 <b>by </b><i>, , </i>;<br/>

<b>thus </b><a NAME="E7:65"/>
 <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font> is <a href="armstrng.html#V5" target="_self">(F4)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_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 the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font>, <font color="Maroon">A'</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font>, <font color="Maroon">B'</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</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><a NAME="E1:65_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>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font> &amp; <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#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font> )
 ;<br/>

<a NAME="E2:65_3"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A</font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">B</font>,<font color="Maroon">D</font> &amp; <font color="Maroon">A'</font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">B'</font>,<font color="Maroon">D</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:65_3"/>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font>,<font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="armstrng.html#U3" target="_self">Relationship</a> of <font color="Maroon">D</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="armstrng.html#U3" target="_self">Relationship</a> of <font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D8" target="_self">ARMSTRNG:def 8</a><br/>

<b>assume </b><a NAME="E1:65_3_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font></span>)</span>
 ;<br/>

<a NAME="E2:65_3_1"/><i><font color="Green">E45</font></i>: 
( <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">A'</font> &amp; <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">A'</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font> &amp; <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>then </b><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <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>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="txt" href="armstrng.html#E3:65"><i><font color="Green">E40</font></i></a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>
;<br/>
<a NAME="E4:65_3_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:65"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <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>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:65"><i><font color="Green">E40</font></i></a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A'</font>
<b>by </b><i>, <a class="txt" href="armstrng.html#E3:65"><i><font color="Green">E40</font></i></a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>
;<br/>
<a NAME="E6:65_3_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B'</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:65"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B'</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T107">RELAT_1:107</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="relat_1.html#T107">RELAT_1:107</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:65_3"/>
<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>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font>
 ;<br/>


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


</div>
