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

<b>let </b><font color="Maroon">X</font> be    <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>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>set </b><font color="Maroon">DOX</font> =  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">X</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_1"/>
<font color="Maroon">x</font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">y</font>,<font color="Maroon">F</font>
 ;<br/><a NAME="E2:70_1"/><b>then </b>
<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#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:70_1"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E3:70_1"/><b>then </b><i><font color="Green">E33</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> <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="E4:70_1"/><b>then </b><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="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>thus </b><a NAME="E5:70_1"/>
<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>by </b><i><a class="txt" href="armstrng.html#E7:70_1"><i><font color="Green">E38</font></i></a></i>;<br/><b>given </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><a NAME="E6:70_1"/><i><font color="Green">E35</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> &amp; ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x'</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</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="armstrng.html#NR6" target="_self">&lt;=</a> <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> )
 ;<br/><a NAME="E7:70_1"/><i><font color="Green">E38</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#NR2" target="_self">&lt;&gt;</a> <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>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E8:70_1"/>
<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">x</font>,<font color="Maroon">y</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">x'</font>,<font color="Maroon">y'</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="E9:70_1"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E2:70"><i><font color="Green">E39</font></i></a>, , , <a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:70"/><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> <font color="Maroon">F</font> &amp; ( 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>  holds <br/>(  not <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> or ( not <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x'</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y'</font> ) or  not <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="armstrng.html#NR6" target="_self">&lt;=</a> <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> ) ) )
 ;<br/>

<a NAME="E3:70"/>
<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="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="70_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:70_2"/>
<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>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">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:70_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <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> &amp; <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">x</font>,<font color="Maroon">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span>,<font color="Maroon">z</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">x'</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:70_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <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> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E2:70"><i><font color="Green">E39</font></i></a>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x'</font>, <font color="Maroon">y'</font> = <font color="Maroon">y'</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="armstrng.html#E2:70_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E6:70_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x'</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:70"><i><font color="Green">E39</font></i></a>, <a class="txt" href="armstrng.html#E2:70_2"><i><font color="Green">E40</font></i></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="E8:70_2"/><i><font color="Green">E45</font></i>: 
( <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">x</font>,<font color="Maroon">y</font></span><a href="armstrng.html#K6" target="_self">]</a></span>,<font color="Maroon">z</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><a class="txt" href="armstrng.html#E2:70"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E9:70_2"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <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> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E10:70_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="armstrng.html#E2:70"><i><font color="Green">E39</font></i></a>, <a class="txt" href="armstrng.html#E2:70_2"><i><font color="Green">E40</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:70"/><b>then </b>
<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> <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/>
<a NAME="E5:70"/><b>then </b>
<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#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font>
 
;<br/>
<b>hence </b><a NAME="E6:70"/>
<font color="Maroon">x</font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">y</font>,<font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


</div>
