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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">B</font> =  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font>;<br/>
<a NAME="E1:86"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E2:86"/>
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">X</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E3:86"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:86"/>
 <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font> is <a href="armstrng.html#V1" target="_self">(B1)</a>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E5:86"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E6:86"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font>
 ;<br/>

<b>consider </b><font color="Maroon">a'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:86"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E9:86"/><i><font color="Green">E39</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>consider </b><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="E11:86"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E12:86"/><i><font color="Green">E43</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">b'</font> holds <br/><font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b'</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">ab</font> = <font color="Maroon">a'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b'</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:86_1"/><i><font color="Green">E44</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="E2:86_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ab</font>
 ;<br/><a NAME="E3:86_1"/>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b'</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/><a NAME="E4:86_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b'</font> )
 <b>by </b><i><a class="ref" href="armstrng.html#T2" target="_self">Th2</a>, , </i>;<br/><b>hence </b><a NAME="E5:86_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ab</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:86"/>
<font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K11" target="_self">enclosure_of</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


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