<?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">x</font> be   <a href="armstrng.html#NM3">Dependency</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><a NAME="E1:69"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">FF</font> = <font color="Maroon">F</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="armstrng.html#NM2">Dependency-set</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is    <a href="armstrng.html#M1" target="_self">Element</a> of <font color="Maroon">FF</font> : <font color="Maroon">x</font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Olive">y</font> </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<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/>
<a NAME="E4:69"/><i><font color="Green">E34</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/>
<a NAME="E5:69"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:69_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:69_1"/><b>then </b>
 ex <font color="Olive">b</font> being   <a href="armstrng.html#M1" target="_self">Element</a> of <font color="Maroon">FF</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">x</font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Olive">b</font> )
 
;<br/>
<a NAME="E3:69_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> &amp;  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</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> )
 
;<br/>
<b>hence </b><a NAME="E4:69_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <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>
 <b>by </b><i><a class="txt" href="armstrng.html#E9:69"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:69"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">S</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="txt" href="armstrng.html#E9:69"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E7:69"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E9:69"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">m</font> <a href="waybel_4.html#R2">is_maximal_wrt</a> <font color="Maroon">S</font>, <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E12:69"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<a NAME="E10:69"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E14:69"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="armstrng.html#M1" target="_self">Element</a> of <font color="Maroon">FF</font><b> such that </b><br/><a NAME="E12:69"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">y</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="E14:69"/><i><font color="Green">E40</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>, </i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>take </b>
<font color="Maroon">b</font>
;<br/>

<a NAME="E15:69"/>
<font color="Maroon">m</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="69_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:69_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><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="E2:69_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E3:69_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E4:69_2"/><i><font color="Green">E45</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#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="E6:69_2"/><i><font color="Green">E46</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>
 <b>and </b><br/><a NAME="E7:69_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">e</font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:69_2"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="armstrng.html#M1" target="_self">Element</a> of <font color="Maroon">FF</font> <b>by </b><i><a class="txt" href="armstrng.html#E2:69_2"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E9:69_2"/>
( <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="txt" href="armstrng.html#E6:69_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E10:69_2"/><b>then </b>
<font color="Maroon">x</font> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:69_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
;<br/>
<b>hence </b><a NAME="E12:69_2"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E14:69"><i><font color="Green">E40</font></i></a>, <a class="txt" href="armstrng.html#E3:69_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="armstrng.html#E4:69_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:69"/><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">F</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:69"/>
<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/>;<br/>

<b>thus </b><a NAME="E18:69"/>
<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> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div>
