<?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">F</font> =  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">X</font> = the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font>;<br/>
<a NAME="E1:105"/><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#V6" target="_self">full_family</a>
 
<b>by </b><i><a class="txt" href="armstrng.html#E5:105"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E2:105"/>
 <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font></span>)</span> is <a href="armstrng.html#V1" target="_self">(B1)</a>
 
<b>by </b><i>, <a class="txt" href="armstrng.html#E17:105"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E3:105"/><b>then </b>
the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">B</font> being   <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> being   <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><b> such that </b><br/><a NAME="E5:105"/><i><font color="Green">E34</font></i>: 
( the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">B</font>, <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E16:105"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E6:105"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A</font>,the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font></span><a href="tarski.html#K4">]</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"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="armstrng.html#E9:105"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E7:105"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K13" target="_self">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:105"/>
 <a href="armstrng.html#K13" target="_self">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font></span>)</span> is <a href="xboole_0.html#V1">(C1)</a>
 ;<br/>

<b>let </b><font color="Maroon">k1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">k2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D28" target="_self">ARMSTRNG:def 28</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E9:105"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K13" target="_self">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font></span>)</span>
 <b>and </b><br/><a NAME="E10:105"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K13" target="_self">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font></span>)</span>
 <b>and </b><br/><a NAME="E11:105"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">k1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k2</font>
 ;<br/>

<b>consider </b><font color="Maroon">a1</font> being   <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><b> such that </b><br/><a NAME="E13:105"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 <b>and </b><br/><a NAME="E14:105"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font></span><a href="tarski.html#K4">]</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"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a2</font> being   <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><b> such that </b><br/><a NAME="E16:105"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 <b>and </b><br/><a NAME="E17:105"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a2</font>,the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font></span><a href="tarski.html#K4">]</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"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E18:105"/><i><font color="Green">E46</font></i>: 
 <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">D</font></span>)</span> is <a href="armstrng.html#V9" target="_self">(M2)</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:105"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a1</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font></span>)</span></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">a2</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">D</font></span>)</span></span><a href="armstrng.html#K6" target="_self">]</a></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/>
<b>hence </b><a NAME="E20:105"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</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>, , , <a class="txt" href="armstrng.html#E11:105"><i><font color="Green">E39</font></i></a></i>;<br/>


</div>
