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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be   <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E62">A1</font></i>: 
 <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c3">G</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">DCF</font> =  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>;<br/>
<b>set </b><font color="Maroon" title="c5">DCG</font> =  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:115_1_1"/><i><font color="Green" title="E63">A2</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:115_1_1"/><i><font color="Green" title="E64">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T10" target="_self" title="ARMSTRNG:th.10">Th10</a></i>;<br/><a NAME="E4:115_1_1"/>
for <font color="Olive" title="b1">c</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>  st <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> &amp; not <font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:115"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="armstrng.html#E1:115_1_1"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="armstrng.html#E3:115_1_1"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="armstrng.html#T59" target="_self" title="ARMSTRNG:th.59">Th59</a></i>;<br/><b>hence </b><a NAME="E5:115_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:115_1_1"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="armstrng.html#T59" target="_self" title="ARMSTRNG:th.59">Th59</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:115_1"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E4:115_1"/><i><font color="Green" title="E64">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T10" target="_self" title="ARMSTRNG:th.10">Th10</a></i>;<br/><a NAME="E5:115_1"/>
for <font color="Olive" title="b1">c</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>  st <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> &amp; not <font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:115"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="armstrng.html#E2:115_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="armstrng.html#E4:115_1"><i><font color="Green" title="E64">A5</font></i></a>, <a class="ref" href="armstrng.html#T59" target="_self" title="ARMSTRNG:th.59">Th59</a></i>;<br/><b>hence </b><a NAME="E6:115_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:115_1"><i><font color="Green" title="E64">A5</font></i></a>, <a class="ref" href="armstrng.html#T59" target="_self" title="ARMSTRNG:th.59">Th59</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:115"/>
 <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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