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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">B</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<b>set </b><font color="Maroon">F</font> = <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font>;<br/>
<b>set </b><font color="Maroon">ss</font> =  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font> as   <a href="armstrng.html#NM5">Full-family</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F'</font>;<br/>
<a NAME="E2:83"/><i><font color="Green">E33</font></i>: 
 <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F'</font> is <a href="armstrng.html#V8" target="_self">(M1)</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:83"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/>
<b>reconsider </b><font color="Maroon">x''</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <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="E7:83"/><i><font color="Green">E35</font></i>: 
<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> <a href="armstrng.html#R3" target="_self">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">x''</font>,<font color="Maroon">x''</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E8:83"/><i><font color="Green">E38</font></i>: 
<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> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F'</font>
 <b>by </b><i>, <a class="txt" href="armstrng.html#E8:83"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E9:83"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x''</font> &amp; <font color="Maroon">x''</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:83"/>
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>then 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="E12:83"/><i><font color="Green">E40</font></i>: 
<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> <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="E13:83"/><i><font color="Green">E43</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> <font color="Maroon">B</font> &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>
 ;<br/>
<a NAME="E14:83"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b'</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="E15:83"/><b>then </b>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, , <a class="ref" href="armstrng.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E16:83"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E17:83"/>
<font color="Maroon">a</font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">b</font>,<font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , <a class="txt" href="armstrng.html#E7:83"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:83"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i/>;<br/>


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