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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="armstrng.html#NM2">Dependency-set</a> of <font color="Maroon">X</font>;<br/>



<b>set </b><font color="Maroon">dcF</font> =  <a href="armstrng.html#K12" target="_self">Dependency-closure</a> <font color="Maroon">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">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:116_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K14" target="_self">charact_set</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:116_1_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:116_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><a NAME="E5:116_1_1"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="armstrng.html#K12" target="_self">Dependency-closure</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><a NAME="E6:116_1_1"/><b>then </b>
( <font color="Maroon">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K12" target="_self">Dependency-closure</a> <font color="Maroon">F</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>, </i>;<br/><b>hence </b><a NAME="E7:116_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K14" target="_self">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" target="_self">Dependency-closure</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:116_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K14" target="_self">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" target="_self">Dependency-closure</a> <font color="Maroon">F</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:116_1"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K12" target="_self">Dependency-closure</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:116_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><a NAME="E6:116_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D2" target="_self">Def2</a>, </i>;<br/><b>assume </b><a NAME="E7:116_1"/>
not <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K14" target="_self">charact_set</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E8:116_1"/><b>then </b><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span> ;<br/><a NAME="E9:116_1"/>
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E10:116_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="armstrng.html#K12" target="_self">Dependency-closure</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">b</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E12:116_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E13:116_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span>  &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D3" target="_self">Def3</a>, </i>;<br/><a NAME="E14:116_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E15:116_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">c</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E16:116_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:116"/>
 <a href="armstrng.html#K14" target="_self">charact_set</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K14" target="_self">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" target="_self">Dependency-closure</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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