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<span class="kw">let </span><font color="Maroon" title="c1">D</font> be    <a href="armstrng.html#L1" title="ARMSTRNG:struct.1">DB-Rel</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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="armstrng.html#NV15" title="ARMSTRNG:NV.15">(C1)</a>  &amp;  <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="armstrng.html#V11" title="ARMSTRNG:attr.11">without_proper_subsets</a>  )</span><br/>

<span class="kw">set </span><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/>
<span class="kw">set </span><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="E1:106"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T25" target="_self" title="ARMSTRNG:th.25">Th25</a></span>;<br/>
<a NAME="E2:106"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A2</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T30" target="_self" title="ARMSTRNG:th.30">Th30</a></span>;<br/>
<a NAME="E3:106"/>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="armstrng.html#T34" target="_self" title="ARMSTRNG:th.34">Th34</a></span>;<br/>
<a NAME="E4:106"/><span class="kw">then </span>
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#D4" target="_self" title="ARMSTRNG:def.4">Def4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:106"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
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>
 <span class="kw">and </span><br/><a NAME="E7:106"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T33" target="_self" title="ARMSTRNG:th.33">Th33</a></span>;<br/>
<a NAME="E8:106"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E7:106"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></span>;<br/>
<a NAME="E9:106"/><span class="kw">then </span>
<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/>
<span class="kw">hence </span><a NAME="E10:106"/>
 <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="armstrng.html#NV15" title="ARMSTRNG:NV.15">(C1)</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="armstrng.html#V11" title="ARMSTRNG:attr.11">without_proper_subsets</a> </span><br/>

<span class="kw">let </span><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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="armstrng.html#D28" target="_self" title="ARMSTRNG:def.28">ARMSTRNG:def 28</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> &amp; <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> implies <font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E11:106"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E12:106"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E13:106"/><span class="lab"><font color="Green" title="E62">A7</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">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><span class="kw"> such that </span><br/><a NAME="E15:106"/><span class="lab"><font color="Green" title="E63">A8</font></span>: 
<font color="Maroon" title="c7">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a2</font>
 <span class="kw">and </span><br/><a NAME="E16:106"/><span class="lab"><font color="Green" title="E64">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:106"><span class="lab"><font color="Green" title="E61">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">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><span class="kw"> such that </span><br/><a NAME="E18:106"/><span class="lab"><font color="Green" title="E65">A10</font></span>: 
<font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1</font>
 <span class="kw">and </span><br/><a NAME="E19:106"/><span class="lab"><font color="Green" title="E66">A11</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:106"><span class="lab"><font color="Green" title="E60">A5</font></span></a></span>;<br/>
<a NAME="E20:106"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">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="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:106"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E18:106"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="txt" href="#E15:106"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:106"/>
<font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:106"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="txt" href="#E19:106"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E15:106"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="txt" href="#E16:106"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="txt" href="#E2:106"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="armstrng.html#D20" target="_self" title="ARMSTRNG:def.20">Def20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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