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

<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#V6" target="_self">full_family</a> <a href="armstrng.html#NM2">Dependency-set</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/>
<b>set </b><font color="Maroon">MEF</font> =  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font>;<br/>
<b>thus </b><a NAME="E1:72"/>
 <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_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#D19" target="_self">ARMSTRNG:def 19</a><br/>

<a NAME="E1:72_1"/><i><font color="Green">E33</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">A</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="E3:72_1"/><i><font color="Green">E34</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">F</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="E4:72_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/>;<br/>
<a NAME="E5:72_1"/><b>then </b><i><font color="Green">E35</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><a class="txt" href="armstrng.html#E16:72_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E6:72_1"/>
<font color="Maroon">MA</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_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="E1:72_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="E2:72_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E16:72_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:72_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">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">MA</font><b> such that </b><br/><a NAME="E9:72_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</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="E10:72_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</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="E11:72_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E16:72_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">A'</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">B'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:72_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">A'</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">B'</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">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A'</font>,<font color="Maroon">B'</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A'</font>, <font color="Maroon">B'</font> = <font color="Maroon">B'</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">A'</font>
;<br/>

<b>take </b>
<font color="Maroon">B'</font>
;<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="E16:72_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</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">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> )
 <b>by </b><i><a class="txt" href="armstrng.html#E16:72_1"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E17:72_1"/>
<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>
 <b>by </b><i>, </i>;<br/>

<a NAME="E18:72_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="72_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:72_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E16:72_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="armstrng.html#E5:72_1_2"><i><font color="Green">E46</font></i></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">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:72_1_2"/><i><font color="Green">E44</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> <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">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">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="E4:72_1_2"/><i><font color="Green">E45</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<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/>;<br/>
<a NAME="E5:72_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">z</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="E6:72_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">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:72_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MA</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E16:72_1"><i><font color="Green">E43</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E8:72_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E4:72_1_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="E19:72_1"/><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="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>, </i>;<br/>
<b>hence </b><a NAME="E20:72_1"/>
<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>
 ;<br/>


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

<b>thus </b><a NAME="E2:72"/>
 <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font> is <a href="armstrng.html#V9" target="_self">(M2)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><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#D20" target="_self">ARMSTRNG:def 20</a><br/>

<b>assume </b><a NAME="E1:72_2"/><i><font color="Green">E50</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> &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>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</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/>

<a NAME="E2:72_2"/><i><font color="Green">E74</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="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>, </i>;<br/>
<b>assume </b><a NAME="E3:72_2"/>
(  not <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> or  not <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 ;<br/>

<a NAME="E4:72_2"/><b>then </b><i><font color="Green">E75</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#NR2" target="_self">&lt;&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="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:72_2"/>
<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="E6:72_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>

<b>thus </b><a NAME="E3:72"/>
 <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font> is <a href="armstrng.html#V10" target="_self">(M3)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_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#D21" target="_self">ARMSTRNG:def 21</a><br/>

<b>assume </b><a NAME="E1:72_3"/><i><font color="Green">E79</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> &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>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font> &amp; <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E2:72_3"/><b>then </b><i><font color="Green">E80</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">B</font>,<font color="Maroon">B'</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:72_3"/>
<font color="Maroon">A'</font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">B'</font>,<font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:72_3"/><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="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:72_3"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">B</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>, <a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E6:72_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
;<br/>
<a NAME="E7:72_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">A</font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">B</font>,<font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:72_3"/><b>then </b><i><font color="Green">E84</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/>;<br/>
<a NAME="E9:72_3"/><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="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:72_3"/><b>then </b><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>,<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/>
<a NAME="E11:72_3"/>
<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#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E12:72_3"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A</font>,<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="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/>;<br/>
<a NAME="E13:72_3"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E14:72_3"/>
<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#T11">XBOOLE_1:11</a></i>;<br/>


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


</div>
