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<b>let </b><font color="Maroon" title="c1">X</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> be   <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>set </b><font color="Maroon" title="c3">dcF</font> =  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">A</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="E1:116_1_1"/><i><font color="Green" title="E63">A1</font></i>: 
<font color="Maroon" title="c4">A</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>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:116_1_1"/><i><font color="Green" title="E64">A2</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E4:116_1_1"/><i><font color="Green" title="E65">A3</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> &amp; not <font color="Maroon" title="c6">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> )
 <b>by </b><i><a class="ref" href="armstrng.html#T57" target="_self" title="ARMSTRNG:th.57">Th57</a></i>;<br/><a NAME="E5:116_1_1"/>
<font color="Maroon" title="c2">F</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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="ref" href="armstrng.html#D25" target="_self" title="ARMSTRNG:def.25">Def25</a></i>;<br/><a NAME="E6:116_1_1"/><b>then </b>
( <font color="Maroon" title="c4">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <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#E1:116_1_1"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="armstrng.html#E3:116_1_1"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="armstrng.html#T57" target="_self" title="ARMSTRNG:th.57">Th57</a></i>;<br/><b>hence </b><a NAME="E7:116_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:116_1_1"><i><font color="Green" title="E65">A3</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:116_1"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:116_1"/><i><font color="Green" title="E64">A5</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <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>and </b><br/><a NAME="E5:116_1"/><i><font color="Green" title="E65">A6</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> &amp; not <font color="Maroon" title="c6">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> )
 <b>by </b><i><a class="ref" href="armstrng.html#T57" target="_self" title="ARMSTRNG:th.57">Th57</a></i>;<br/><a NAME="E6:116_1"/><i><font color="Green" title="E66">A7</font></i>: 
<font color="Maroon" title="c4">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:116_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="armstrng.html#T57" target="_self" title="ARMSTRNG:th.57">Th57</a></i>;<br/><b>assume </b><a NAME="E7:116_1"/>
not <font color="Maroon" title="c4">A</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>
 ;<br/><a NAME="E8:116_1"/><b>then </b><i><font color="Green" title="E67">A8</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b2">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:116_1"><i><font color="Green" title="E66">A7</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c7">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">x</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="b3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> </span>}</span> ;<br/><a NAME="E9:116_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">x</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="b3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font>
 ;<br/><a NAME="E10:116_1"/><b>then </b>
 <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> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">x</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="b3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="armstrng.html#T40" target="_self" title="ARMSTRNG:th.40">Th40</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E12:116_1"/><i><font color="Green" title="E68">A9</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <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="c8">a</font>,<font color="Maroon" title="c9">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E13:116_1"/><i><font color="Green" title="E69">A10</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">c</font> where <font color="Olive" title="b2">c</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">c</font> holds <br/><font color="Olive" title="b4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">c</font> </span>}</span>  &amp; <font color="Maroon" title="c8">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c9">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:116_1"><i><font color="Green" title="E64">A5</font></i></a></i>;<br/><a NAME="E14:116_1"/><i><font color="Green" title="E70">A11</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E12:116_1"><i><font color="Green" title="E68">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E15:116_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">x</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="b3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E6:116_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="txt" href="armstrng.html#E8:116_1"><i><font color="Green" title="E67">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E16:116_1"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E5:116_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="armstrng.html#E13:116_1"><i><font color="Green" title="E69">A10</font></i></a>, <a class="txt" href="armstrng.html#E14:116_1"><i><font color="Green" title="E70">A11</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:116"/>
 <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> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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