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<b>let </b><font color="Maroon" title="c1">D</font> be    <a href="armstrng.html#L1" title="ARMSTRNG:struct.1">DB-Rel</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">F</font> =  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font>;<br/>
<b>set </b><font color="Maroon" title="c3">X</font> = the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font>;<br/>
<a NAME="E1:105"/><i><font color="Green" title="E58">A1</font></i>: 
 <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a>
 
<b>by </b><i><a class="ref" href="armstrng.html#T25" target="_self" title="ARMSTRNG:th.25">Th25</a></i>;<br/>
<a NAME="E2:105"/><b>then </b>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span>)</span> is <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a>
 
<b>by </b><i><a class="ref" href="armstrng.html#T34" target="_self" title="ARMSTRNG:th.34">Th34</a></i>;<br/>
<a NAME="E3:105"/><b>then </b>
the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="armstrng.html#D4" target="_self" title="ARMSTRNG:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E5:105"/><i><font color="Green" title="E59">A2</font></i>: 
( the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c5">A</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c4">B</font>, <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="ref" href="armstrng.html#T33" target="_self" title="ARMSTRNG:th.33">Th33</a></i>;<br/>
<a NAME="E6:105"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="armstrng.html#E5:105"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/>
<a NAME="E7:105"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:105"/>
 <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">k1</font>, <font color="Maroon" title="c7">k2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D28" target="_self" title="ARMSTRNG:def.28">ARMSTRNG:def 28</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E9:105"/><i><font color="Green" title="E60">A3</font></i>: 
<font color="Maroon" title="c6">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span>)</span>
 <b>and </b><br/><a NAME="E10:105"/><i><font color="Green" title="E61">A4</font></i>: 
<font color="Maroon" title="c7">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span>)</span>
 <b>and </b><br/><a NAME="E11:105"/><i><font color="Green" title="E62">A5</font></i>: 
<font color="Maroon" title="c6">k1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">k2</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">a1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E13:105"/><i><font color="Green" title="E63">A6</font></i>: 
<font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font>
 <b>and </b><br/><a NAME="E14:105"/><i><font color="Green" title="E64">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">a1</font>,the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E9:105"><i><font color="Green" title="E60">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">a2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E16:105"/><i><font color="Green" title="E65">A8</font></i>: 
<font color="Maroon" title="c7">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a2</font>
 <b>and </b><br/><a NAME="E17:105"/><i><font color="Green" title="E66">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a2</font>,the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E10:105"><i><font color="Green" title="E61">A4</font></i></a></i>;<br/>
<a NAME="E18:105"/><i><font color="Green" title="E67">A10</font></i>: 
 <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span>)</span> is <a href="armstrng.html#V9" title="ARMSTRNG:attr.9">(M2)</a>
 
<b>by </b><i><a class="txt" href="armstrng.html#E1:105"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="armstrng.html#T30" target="_self" title="ARMSTRNG:th.30">Th30</a></i>;<br/>
<a NAME="E19:105"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a1</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a2</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="armstrng.html#E11:105"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="armstrng.html#E13:105"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="armstrng.html#E16:105"><i><font color="Green" title="E65">A8</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E20:105"/>
<font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k2</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E13:105"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="armstrng.html#E14:105"><i><font color="Green" title="E64">A7</font></i></a>, <a class="txt" href="armstrng.html#E16:105"><i><font color="Green" title="E65">A8</font></i></a>, <a class="txt" href="armstrng.html#E17:105"><i><font color="Green" title="E66">A9</font></i></a>, <a class="txt" href="armstrng.html#E18:105"><i><font color="Green" title="E67">A10</font></i></a>, <a class="ref" href="armstrng.html#D20" target="_self" title="ARMSTRNG:def.20">Def20</a></i>;<br/>


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