<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M</font> be   <a href="armstrng.html#NM2">Dependency-set</a> of <font color="Maroon">X</font>;<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>assume </b><b>that </b><br/><a NAME="E1:73"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">M</font> is <a href="armstrng.html#V8" target="_self">(M1)</a> &amp; <font color="Maroon">M</font> is <a href="armstrng.html#V9" target="_self">(M2)</a> &amp; <font color="Maroon">M</font> is <a href="armstrng.html#V10" target="_self">(M3)</a> )
 <b>and </b><br/><a NAME="E2:73"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">B</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">A'</font>, <font color="Olive">B'</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A'</font>,<font color="Olive">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="Olive">A</font>,<font color="Olive">B</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A'</font>,<font color="Olive">B'</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> ) </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon">DOX</font> =  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font>;<br/>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:73_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<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="E3:73_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <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="E4:73_1"/><i><font color="Green">E39</font></i>: 
 ex <font color="Olive">a'</font>, <font color="Olive">b'</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">a'</font>,<font color="Olive">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">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">a'</font>,<font color="Olive">b'</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</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="E6:73_1"/><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="armstrng.html#R3" target="_self">&gt;=</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> &amp; <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">M</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">b</font>;<br/><b>take </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font>;<br/><b>take </b><font color="Maroon">b'</font> = <font color="Maroon">b'</font>;<br/><b>thus </b><a NAME="E7:73_1"/>
( <font color="Maroon">x</font> <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> &amp; <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">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <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">M</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:73_2"/>
<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">F</font>
 ;<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>, <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="E3:73_2"/><i><font color="Green">E44</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="E4:73_2"/><i><font color="Green">E45</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">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <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">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E4:73_2"><i><font color="Green">E45</font></i></a></i>;<br/><b>take </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font>;<br/><b>take </b><font color="Maroon">b'</font> = <font color="Maroon">b'</font>;<br/><b>thus </b><a NAME="E5:73_2"/>
( <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">A</font>,<font color="Maroon">B</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <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">M</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:73_3_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<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="E3:73_3_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <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>by </b><i/>;<br/><a NAME="E4:73_3_1"/>
<font color="Maroon">x</font> <a href="waybel_4.html#R2">is_maximal_wrt</a> <font color="Maroon">F</font>, <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:73_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, , </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a><br/>

<b>given </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:73_3_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font> )
 ;<br/>

<b>consider </b><font color="Maroon">e</font> being   <a href="armstrng.html#NM3">Dependency</a> of <font color="Maroon">X</font>, <font color="Maroon">f</font> being   <a href="armstrng.html#NM3">Dependency</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:73_3_1_1"/><i><font color="Green">E75</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">e</font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:73_3_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E3:73_3_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">d</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">c'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">d'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:73_3_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E8:73_3_1_1"/><i><font color="Green">E81</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</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">c</font>,<font color="Maroon">d</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:73_3_1_1"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</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">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 
<b>by </b><i>, <a class="txt" href="armstrng.html#E3:73_3_1"><i><font color="Green">E50</font></i></a>, , , , </i>;<br/>
<a NAME="E10:73_3_1_1"/><b>then </b>
( <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:73"><i><font color="Green">E43</font></i></a>, , , , </i>;<br/>
<b>hence </b><a NAME="E11:73_3_1_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="armstrng.html#E3:73_3_1"><i><font color="Green">E50</font></i></a>, , , , </i>;<br/>


</div><b>end;</b></div><a NAME="E5:73_3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font></span>)</span> <a href="armstrng.html#K1" target="_self">Maximal_in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:73_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font>
 ;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:73_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E3:73_3"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font></span>)</span> <a href="armstrng.html#K1" target="_self">Maximal_in</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E4:73_3"/><b>then </b><i><font color="Green">E83</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="waybel_4.html#R2">is_maximal_wrt</a> <font color="Maroon">F</font>, <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <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>, <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="E6:73_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <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="E7:73_3"/><i><font color="Green">E85</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">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <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">M</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E8:73_3"/><i><font color="Green">E109</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> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/><b>assume </b><a NAME="E9:73_3"/><i><font color="Green">E110</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E10:73_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><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>,<span class="p2"><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></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E11:73_3"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:73"/>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<b>thus </b><a NAME="E7:73"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V6" target="_self">full_family</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:73_4"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V3" target="_self">(F1)</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D15" target="_self">ARMSTRNG:def 15</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D12" target="_self">ARMSTRNG:def 12</a><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="E2:73_4_1"/><i><font color="Green">E111</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">A</font>,<font color="Maroon">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <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">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E4:73"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E3:73_4_1"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:73_4_2"/><i><font color="Green">E112</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> <font color="Maroon">F</font> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<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="E3:73_4_2"/><i><font color="Green">E113</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">A</font>,<font color="Maroon">B</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <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">M</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">B1'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:73_4_2"/><i><font color="Green">E114</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">B1'</font>,<font color="Maroon">C'</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">B</font>,<font color="Maroon">C</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">B1'</font>,<font color="Maroon">C'</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:73_4_2"/>
( <font color="Maroon">B1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B'</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E7:73_4_2"/><b>then </b>
<font color="Maroon">B1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:73_4_2"/><b>then </b>
( <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">C'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B'</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E4:73"><i><font color="Green">E43</font></i></a>, , , </i>;<br/><a NAME="E9:73_4_2"/><b>then </b><i><font color="Green">E115</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">A'</font>,<font color="Maroon">C'</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E10:73_4_2"/>
( <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C'</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E11:73_4_2"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A'</font>,<font color="Maroon">C'</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">A</font>,<font color="Maroon">C</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E12:73_4_2"/><b>then </b>
<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">A</font>,<font color="Maroon">C</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E13:73_4_2"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">C</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:73_4"/>
<font color="Maroon">F</font> is <a href="relat_2.html#V8">(F2)</a>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E4:73_4"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V4" target="_self">(F3)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">A'</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B'</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D13" target="_self">ARMSTRNG:def 13</a><br/>

<b>assume </b><a NAME="E1:73_4_3"/><i><font color="Green">E116</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> <font color="Maroon">F</font> &amp; <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">A'</font>,<font color="Maroon">B'</font></span><a href="armstrng.html#K6" target="_self">]</a></span> )
 ;<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="E3:73_4_3"/><i><font color="Green">E117</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">A</font>,<font color="Maroon">B</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <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">M</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E4:73_4_3"/>
<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">A'</font>,<font color="Maroon">B'</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:73_4_3"/>
<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">F</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:73_4"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V5" target="_self">(F4)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">A'</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B'</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D14" target="_self">ARMSTRNG:def 14</a><br/>

<b>assume </b><a NAME="E1:73_4_4"/><i><font color="Green">E118</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> <font color="Maroon">F</font> &amp; <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">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>, <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="E3:73_4_4"/><i><font color="Green">E119</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"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font></span>)</span></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <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">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E4:73"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<a NAME="E4:73_4_4"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">A''</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B''</font> )
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">b1</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:73_4_4"/><i><font color="Green">E121</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</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">A</font>,<font color="Maroon">B</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:73_4_4"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">a1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:73_4_4"/><b>then </b>
<font color="Maroon">a1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T10">XBOOLE_1:10</a></i>;<br/>
<a NAME="E9:73_4_4"/><b>then </b>
<font color="Maroon">a1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B''</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:73_4_4"/><b>then </b>
<font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B''</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:73"><i><font color="Green">E43</font></i></a>, , , </i>;<br/>
<a NAME="E11:73_4_4"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B''</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">a1'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">b1'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E13:73_4_4"/><i><font color="Green">E124</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</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">A'</font>,<font color="Maroon">B'</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E14:73_4_4"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">a1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b1'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:73_4_4"/><b>then </b>
<font color="Maroon">a1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T10">XBOOLE_1:10</a></i>;<br/>
<a NAME="E16:73_4_4"/><b>then </b>
<font color="Maroon">a1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B''</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E17:73_4_4"/><b>then </b>
<font color="Maroon">b1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B''</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:73"><i><font color="Green">E43</font></i></a>, , , </i>;<br/>
<a NAME="E18:73_4_4"/><b>then </b>
<font color="Maroon">B'</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>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:73_4_4"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B''</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B''</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E20:73_4_4"/><b>then </b>
<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"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font></span>)</span></span><a href="armstrng.html#K6" target="_self">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E21:73_4_4"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font></span>)</span></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">G</font> be   <a href="armstrng.html#NM5">Full-family</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E8:73"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:73_5_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<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="E3:73_5_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">x</font> <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>by </b><i/>;<br/><a NAME="E4:73_5_1"/><i><font color="Green">E129</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">y</font> being  <a href="armstrng.html#NM3">Dependency</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <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">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span> );<br/><b>consider </b><font color="Maroon">MA</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:73_5_1"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MA</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><a NAME="E7:73_5_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">MA</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , </i>;<br/><a NAME="E8:73_5_1"/>
<font color="Maroon">MA</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5_1_1"><b>proof </b></a><div class="add">

<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="E1:73_5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MA</font>
 ;<br/>

<b>hence </b><a NAME="E2:73_5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E9:73_5_1"/><b>then </b>
<font color="Maroon">MA</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <span class="p2">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">MA</font><b> such that </b><br/><a NAME="E11:73_5_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">m</font> <a href="waybel_4.html#R2">is_maximal_wrt</a> <font color="Maroon">MA</font>, <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:73_5_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MA</font>
 <b>by </b><i>, <a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a></i>;<br/><a NAME="E13:73_5_1"/>
<font color="Maroon">m</font> <a href="waybel_4.html#R2">is_maximal_wrt</a> <font color="Maroon">G</font>, <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:73_5_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="armstrng.html#T2" target="_self">Th2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a><br/>

<b>given </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:73_5_1_2"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font> )
 ;<br/>

<b>consider </b><font color="Maroon">mm</font> being   <a href="armstrng.html#NM3">Dependency</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:73_5_1_2"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</font> &amp; <font color="Maroon">mm</font> <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">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span> )
 <b>by </b><i>, <a class="ref" href="armstrng.html#T2" target="_self">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being   <a href="armstrng.html#NM3">Dependency</a> of <font color="Maroon">X</font>, <font color="Maroon">f</font> being   <a href="armstrng.html#NM3">Dependency</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:73_5_1_2"/><i><font color="Green">E136</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">e</font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:73_5_1_2"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:73_5_1_2"/><b>then </b>
<font color="Maroon">f</font> <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">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:73_5_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MA</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:73_5_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:73_5_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font></span>)</span> <a href="armstrng.html#K1" target="_self">Maximal_in</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E15:73_5_1"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">y</font> being   <a href="armstrng.html#NM3">Dependency</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E17:73_5_1"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <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">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span> )
 <b>by </b><i>, <a class="ref" href="armstrng.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E18:73_5_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="armstrng.html#T2" target="_self">Th2</a></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="E20:73_5_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">m</font> <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>by </b><i/>;<br/><b>thus </b><a NAME="E21:73_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, , , , </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:73_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<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>, <font color="Maroon">a1</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">b1</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:73_5"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">x</font> <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="E5:73_5"/><i><font color="Green">E142</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</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">a</font>,<font color="Maroon">b</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i/>;<br/><a NAME="E6:73_5"/>
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:73_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma26</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:73"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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