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<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">F</font> be   <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:73"/><i><font color="Green" title="E43">A1</font></i>: 
( <font color="Maroon" title="c2">M</font> is <a href="armstrng.html#V8" title="ARMSTRNG:attr.8">(M1)</a> &amp; <font color="Maroon" title="c2">M</font> is <a href="armstrng.html#V9" title="ARMSTRNG:attr.9">(M2)</a> &amp; <font color="Maroon" title="c2">M</font> is <a href="armstrng.html#V10" title="ARMSTRNG:attr.10">(M3)</a> )
 <b>and </b><br/><a NAME="E2:73"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b3">A'</font>, <font color="Olive" title="b4">B'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">A'</font>,<font color="Olive" title="b4">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">A'</font>,<font color="Olive" title="b4">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> ) </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon" title="c4">DOX</font> =  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>;<br/>
<div><i><font color="Green" title="E45">A3</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" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:73_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:73_1"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E4:73_1"/><i><font color="Green" title="E47">A5</font></i>: 
 ex <font color="Olive" title="b1">a'</font>, <font color="Olive" title="b2">b'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">a'</font>,<font color="Olive" title="b2">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">a'</font>,<font color="Olive" title="b2">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E2:73"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">a'</font>, <font color="Maroon" title="c9">b'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:73_1"/><i><font color="Green" title="E48">A6</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E4:73_1"><i><font color="Green" title="E47">A5</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c6">a</font>;<br/><b>take </b><font color="Maroon" title="c11">b</font> = <font color="Maroon" title="c7">b</font>;<br/><b>take </b><font color="Maroon" title="c12">a'</font> = <font color="Maroon" title="c8">a'</font>;<br/><b>take </b><font color="Maroon" title="c13">b'</font> = <font color="Maroon" title="c9">b'</font>;<br/><b>thus </b><a NAME="E7:73_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="armstrng.html#E6:73_1"><i><font color="Green" title="E48">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E46">A7</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" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:73_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">b'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:73_2"/><i><font color="Green" title="E47">A8</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E4:73_2"/><i><font color="Green" title="E48">A9</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73"><i><font color="Green" title="E45">A3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c11">a'</font> = <font color="Maroon" title="c9">a'</font>;<br/><b>take </b><font color="Maroon" title="c12">b'</font> = <font color="Maroon" title="c10">b'</font>;<br/><b>thus </b><a NAME="E5:73_2"/>
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a'</font>,<font color="Maroon" title="c12">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a'</font>,<font color="Maroon" title="c12">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73_2"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="armstrng.html#E4:73_2"><i><font color="Green" title="E48">A9</font></i></a></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" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:73_3_1"/><i><font color="Green" title="E47">A10</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:73_3_1"/><i><font color="Green" title="E48">A11</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T10" target="_self" title="ARMSTRNG:th.10">Th10</a></i>;<br/><a NAME="E4:73_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c3">F</font>, <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">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" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:73"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="armstrng.html#E1:73_3_1"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="armstrng.html#E3:73_3_1"><i><font color="Green" title="E48">A11</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D24" title="WAYBEL_4:def.24">WAYBEL_4:def 24</a><br/>

<b>given </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:73_3_1_1"/><i><font color="Green" title="E49">A12</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">e</font>, <font color="Maroon" title="c10">f</font> being   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:73_3_1_1"/><i><font color="Green" title="E50">A13</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c10">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">e</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c10">f</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73_3_1_1"><i><font color="Green" title="E49">A12</font></i></a></i>;<br/>
<a NAME="E6:73_3_1_1"/><i><font color="Green" title="E51">A14</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E5:73_3_1_1"><i><font color="Green" title="E50">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">c</font>, <font color="Maroon" title="c12">d</font>, <font color="Maroon" title="c13">c'</font>, <font color="Maroon" title="c14">d'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:73_3_1_1"/><i><font color="Green" title="E52">A15</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E9:73_3_1_1"/><i><font color="Green" title="E53">A16</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">c'</font>,<font color="Maroon" title="c14">d'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">c'</font>,<font color="Maroon" title="c14">d'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="armstrng.html#E3:73_3_1_1"><i><font color="Green" title="E49">A12</font></i></a></i>;<br/>
<a NAME="E10:73_3_1_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">c'</font>,<font color="Maroon" title="c14">d'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="armstrng.html#E3:73_3_1"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="armstrng.html#E5:73_3_1_1"><i><font color="Green" title="E50">A13</font></i></a>, <a class="txt" href="armstrng.html#E6:73_3_1_1"><i><font color="Green" title="E51">A14</font></i></a>, <a class="txt" href="armstrng.html#E8:73_3_1_1"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="armstrng.html#E9:73_3_1_1"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="armstrng.html#T14" target="_self" title="ARMSTRNG:th.14">Th14</a></i>;<br/>
<a NAME="E11:73_3_1_1"/><b>then </b>
( <font color="Maroon" title="c13">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c14">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E1:73"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="armstrng.html#E1:73_3_1"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="armstrng.html#E3:73_3_1"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="armstrng.html#E9:73_3_1_1"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="armstrng.html#D20" target="_self" title="ARMSTRNG:def.20">Def20</a></i>;<br/>
<b>hence </b><a NAME="E12:73_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73_3_1"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="armstrng.html#E3:73_3_1_1"><i><font color="Green" title="E49">A12</font></i></a>, <a class="txt" href="armstrng.html#E5:73_3_1_1"><i><font color="Green" title="E50">A13</font></i></a>, <a class="txt" href="armstrng.html#E6:73_3_1_1"><i><font color="Green" title="E51">A14</font></i></a>, <a class="txt" href="armstrng.html#E8:73_3_1_1"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="armstrng.html#E9:73_3_1_1"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:73_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D1" target="_self" title="ARMSTRNG:def.1">Def1</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:73_3"/><i><font color="Green" title="E47">A17</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c3">F</font>
 ;<br/><a NAME="E3:73_3"/><b>then </b><i><font color="Green" title="E48">A18</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c5">x</font> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c3">F</font>, <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="armstrng.html#D1" target="_self" title="ARMSTRNG:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">a'</font>, <font color="Maroon" title="c9">b'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:73_3"/><i><font color="Green" title="E49">A19</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E6:73_3"/><i><font color="Green" title="E50">A20</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="armstrng.html#E2:73_3"><i><font color="Green" title="E47">A17</font></i></a></i>;<br/><a NAME="E7:73_3"/><i><font color="Green" title="E51">A21</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:73"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="armstrng.html#E6:73_3"><i><font color="Green" title="E50">A20</font></i></a></i>;<br/><b>assume </b><a NAME="E8:73_3"/><i><font color="Green" title="E52">A22</font></i>: 
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 ;<br/><a NAME="E9:73_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>,<span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:73_3"><i><font color="Green" title="E50">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E10:73_3"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73_3"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="armstrng.html#E5:73_3"><i><font color="Green" title="E49">A19</font></i></a>, <a class="txt" href="armstrng.html#E6:73_3"><i><font color="Green" title="E50">A20</font></i></a>, <a class="txt" href="armstrng.html#E7:73_3"><i><font color="Green" title="E51">A21</font></i></a>, <a class="txt" href="armstrng.html#E8:73_3"><i><font color="Green" title="E52">A22</font></i></a>, <a class="ref" href="waybel_4.html#D24" title="WAYBEL_4:def.24">WAYBEL_4:def 24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:73"/>
<font color="Maroon" title="c2">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>

<b>thus </b><a NAME="E7:73"/>
<font color="Maroon" title="c3">F</font> is <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">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" title="c3">F</font> is <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D15" target="_self" title="ARMSTRNG:def.15">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" title="c5">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">ARMSTRNG:def 12</a><br/>

<b>consider </b><font color="Maroon" title="c6">A'</font>, <font color="Maroon" title="c7">B'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:73_4_1"/><i><font color="Green" title="E47">A23</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c7">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c5">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c7">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E1:73"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="armstrng.html#D19" target="_self" title="ARMSTRNG:def.19">Def19</a></i>;<br/>
<b>thus </b><a NAME="E3:73_4_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c5">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:73"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="armstrng.html#E2:73_4_1"><i><font color="Green" title="E47">A23</font></i></a></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" title="c5">A</font>, <font color="Maroon" title="c6">B</font>, <font color="Maroon" title="c7">C</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:73_4_2"/><i><font color="Green" title="E47">A24</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">C</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c8">A'</font>, <font color="Maroon" title="c9">B'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:73_4_2"/><i><font color="Green" title="E48">A25</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c9">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c9">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E4:73"><i><font color="Green" title="E46">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">B1'</font>, <font color="Maroon" title="c11">C'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:73_4_2"/><i><font color="Green" title="E49">A26</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">B1'</font>,<font color="Maroon" title="c11">C'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">C</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">B1'</font>,<font color="Maroon" title="c11">C'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E4:73"><i><font color="Green" title="E46">A7</font></i></a>, <a class="txt" href="armstrng.html#E1:73_4_2"><i><font color="Green" title="E47">A24</font></i></a></i>;<br/><a NAME="E6:73_4_2"/>
( <font color="Maroon" title="c10">B1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B'</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73_4_2"><i><font color="Green" title="E48">A25</font></i></a>, <a class="txt" href="armstrng.html#E5:73_4_2"><i><font color="Green" title="E49">A26</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><a NAME="E7:73_4_2"/><b>then </b>
<font color="Maroon" title="c10">B1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:73_4_2"/><b>then </b>
( <font color="Maroon" title="c8">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A'</font> &amp; <font color="Maroon" title="c11">C'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B'</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E1:73"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="armstrng.html#E3:73_4_2"><i><font color="Green" title="E48">A25</font></i></a>, <a class="txt" href="armstrng.html#E5:73_4_2"><i><font color="Green" title="E49">A26</font></i></a>, <a class="ref" href="armstrng.html#D21" target="_self" title="ARMSTRNG:def.21">Def21</a></i>;<br/><a NAME="E9:73_4_2"/><b>then </b><i><font color="Green" title="E50">A27</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c9">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c11">C'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><a NAME="E10:73_4_2"/>
( <font color="Maroon" title="c8">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c7">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">C'</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73_4_2"><i><font color="Green" title="E48">A25</font></i></a>, <a class="txt" href="armstrng.html#E5:73_4_2"><i><font color="Green" title="E49">A26</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><a NAME="E11:73_4_2"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c11">C'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">C</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><a NAME="E12:73_4_2"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">A'</font>,<font color="Maroon" title="c9">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">C</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="txt" href="armstrng.html#E9:73_4_2"><i><font color="Green" title="E50">A27</font></i></a>, <a class="ref" href="armstrng.html#T14" target="_self" title="ARMSTRNG:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E13:73_4_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">C</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:73"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="armstrng.html#E3:73_4_2"><i><font color="Green" title="E48">A25</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:73_4"/>
<font color="Maroon" title="c3">F</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">(F2)</a>
 <b>by </b><i><a class="ref" href="armstrng.html#T20" target="_self" title="ARMSTRNG:th.20">Th20</a></i>;<br/>

<b>thus </b><a NAME="E4:73_4"/>
<font color="Maroon" title="c3">F</font> is <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(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" title="c5">A</font>, <font color="Maroon" title="c6">B</font>, <font color="Maroon" title="c7">A'</font>, <font color="Maroon" title="c8">B'</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D13" target="_self" title="ARMSTRNG:def.13">ARMSTRNG:def 13</a><br/>

<b>assume </b><a NAME="E1:73_4_3"/><i><font color="Green" title="E47">A28</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c8">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">b'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:73_4_3"/><i><font color="Green" title="E48">A29</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E4:73"><i><font color="Green" title="E46">A7</font></i></a>, <a class="txt" href="armstrng.html#E1:73_4_3"><i><font color="Green" title="E47">A28</font></i></a></i>;<br/>
<a NAME="E4:73_4_3"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c8">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="armstrng.html#E1:73_4_3"><i><font color="Green" title="E47">A28</font></i></a>, <a class="txt" href="armstrng.html#E3:73_4_3"><i><font color="Green" title="E48">A29</font></i></a>, <a class="ref" href="armstrng.html#T14" target="_self" title="ARMSTRNG:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E5:73_4_3"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c8">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:73"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="armstrng.html#E3:73_4_3"><i><font color="Green" title="E48">A29</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E5:73_4"/>
<font color="Maroon" title="c3">F</font> is <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(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" title="c5">A</font>, <font color="Maroon" title="c6">B</font>, <font color="Maroon" title="c7">A'</font>, <font color="Maroon" title="c8">B'</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D14" target="_self" title="ARMSTRNG:def.14">ARMSTRNG:def 14</a><br/>

<b>assume </b><a NAME="E1:73_4_4"/><i><font color="Green" title="E47">A30</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c8">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">A''</font>, <font color="Maroon" title="c10">B''</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:73_4_4"/><i><font color="Green" title="E48">A31</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">A''</font>,<font color="Maroon" title="c10">B''</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">A''</font>,<font color="Maroon" title="c10">B''</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E1:73"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="armstrng.html#D19" target="_self" title="ARMSTRNG:def.19">Def19</a></i>;<br/>
<a NAME="E4:73_4_4"/><i><font color="Green" title="E49">A32</font></i>: 
( <font color="Maroon" title="c9">A''</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font> &amp; <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B''</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E3:73_4_4"><i><font color="Green" title="E48">A31</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">a1</font>, <font color="Maroon" title="c12">b1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:73_4_4"/><i><font color="Green" title="E50">A33</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c12">b1</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c12">b1</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E4:73"><i><font color="Green" title="E46">A7</font></i></a>, <a class="txt" href="armstrng.html#E1:73_4_4"><i><font color="Green" title="E47">A30</font></i></a></i>;<br/>
<a NAME="E7:73_4_4"/><i><font color="Green" title="E51">A34</font></i>: 
( <font color="Maroon" title="c11">a1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">b1</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E6:73_4_4"><i><font color="Green" title="E50">A33</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/>
<a NAME="E8:73_4_4"/><b>then </b>
<font color="Maroon" title="c11">a1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></i>;<br/>
<a NAME="E9:73_4_4"/><b>then </b>
<font color="Maroon" title="c11">a1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B''</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:73_4_4"><i><font color="Green" title="E49">A32</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:73_4_4"/><b>then </b>
<font color="Maroon" title="c12">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B''</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E1:73"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="armstrng.html#E3:73_4_4"><i><font color="Green" title="E48">A31</font></i></a>, <a class="txt" href="armstrng.html#E6:73_4_4"><i><font color="Green" title="E50">A33</font></i></a>, <a class="ref" href="armstrng.html#D21" target="_self" title="ARMSTRNG:def.21">Def21</a></i>;<br/>
<a NAME="E11:73_4_4"/><b>then </b><i><font color="Green" title="E52">A35</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B''</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E7:73_4_4"><i><font color="Green" title="E51">A34</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">a1'</font>, <font color="Maroon" title="c14">b1'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E13:73_4_4"/><i><font color="Green" title="E53">A36</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">a1'</font>,<font color="Maroon" title="c14">b1'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c8">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">a1'</font>,<font color="Maroon" title="c14">b1'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E4:73"><i><font color="Green" title="E46">A7</font></i></a>, <a class="txt" href="armstrng.html#E1:73_4_4"><i><font color="Green" title="E47">A30</font></i></a></i>;<br/>
<a NAME="E14:73_4_4"/><i><font color="Green" title="E54">A37</font></i>: 
( <font color="Maroon" title="c13">a1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A'</font> &amp; <font color="Maroon" title="c8">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">b1'</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E13:73_4_4"><i><font color="Green" title="E53">A36</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/>
<a NAME="E15:73_4_4"/><b>then </b>
<font color="Maroon" title="c13">a1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></i>;<br/>
<a NAME="E16:73_4_4"/><b>then </b>
<font color="Maroon" title="c13">a1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B''</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:73_4_4"><i><font color="Green" title="E49">A32</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E17:73_4_4"/><b>then </b>
<font color="Maroon" title="c14">b1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B''</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E1:73"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="armstrng.html#E3:73_4_4"><i><font color="Green" title="E48">A31</font></i></a>, <a class="txt" href="armstrng.html#E13:73_4_4"><i><font color="Green" title="E53">A36</font></i></a>, <a class="ref" href="armstrng.html#D21" target="_self" title="ARMSTRNG:def.21">Def21</a></i>;<br/>
<a NAME="E18:73_4_4"/><b>then </b>
<font color="Maroon" title="c8">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B''</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E14:73_4_4"><i><font color="Green" title="E54">A37</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:73_4_4"/><b>then </b>
<font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B''</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">B''</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E11:73_4_4"><i><font color="Green" title="E52">A35</font></i></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E20:73_4_4"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">A''</font>,<font color="Maroon" title="c10">B''</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:73_4_4"><i><font color="Green" title="E49">A32</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E21:73_4_4"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:73"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="armstrng.html#E3:73_4_4"><i><font color="Green" title="E48">A31</font></i></a></i>;<br/>


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


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

<b>let </b><font color="Maroon" title="c5">G</font> be   <a href="armstrng.html#NM5" title="ARMSTRNG:NM.5">Full-family</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E8:73"/><i><font color="Green" title="E47">A38</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c5">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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:73_5_1"/><i><font color="Green" title="E48">A39</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:73_5_1"/><i><font color="Green" title="E49">A40</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T10" target="_self" title="ARMSTRNG:th.10">Th10</a></i>;<br/><a NAME="E4:73_5_1"/><i><font color="Green" title="E50">A41</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T19" target="_self" title="ARMSTRNG:th.19">Th19</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">y</font> being  <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> );<br/><b>consider </b><font color="Maroon" title="c9">MA</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:73_5_1"/><i><font color="Green" title="E51">A42</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">MA</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i><a NAME="E7:73_5_1"/><i><font color="Green" title="E52">A43</font></i>: 
<font color="Maroon" title="c9">MA</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E1:73_5_1"><i><font color="Green" title="E48">A39</font></i></a>, <a class="txt" href="armstrng.html#E3:73_5_1"><i><font color="Green" title="E49">A40</font></i></a>, <a class="txt" href="armstrng.html#E6:73_5_1"><i><font color="Green" title="E51">A42</font></i></a></i>;<br/><a NAME="E8:73_5_1"/>
<font color="Maroon" title="c9">MA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">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" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:73_5_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">MA</font>
 ;<br/>

<b>hence </b><a NAME="E2:73_5_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:73_5_1"><i><font color="Green" title="E51">A42</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:73_5_1"/><b>then </b>
<font color="Maroon" title="c9">MA</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p2">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:73_5_1"><i><font color="Green" title="E50">A41</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">MA</font><b> such that </b><br/><a NAME="E11:73_5_1"/><i><font color="Green" title="E53">A44</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c9">MA</font>, <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E7:73_5_1"><i><font color="Green" title="E52">A43</font></i></a>, <a class="ref" href="armstrng.html#T2" target="_self" title="ARMSTRNG:th.2">Th2</a></i>;<br/><a NAME="E12:73_5_1"/><i><font color="Green" title="E54">A45</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">MA</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E11:73_5_1"><i><font color="Green" title="E53">A44</font></i></a>, <a class="ref" href="waybel_4.html#D24" title="WAYBEL_4:def.24">WAYBEL_4:def 24</a></i>;<br/><a NAME="E13:73_5_1"/>
<font color="Maroon" title="c10">m</font> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c5">G</font>, <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">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" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:73_5_1"><i><font color="Green" title="E51">A42</font></i></a>, <a class="txt" href="armstrng.html#E12:73_5_1"><i><font color="Green" title="E54">A45</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D24" title="WAYBEL_4:def.24">WAYBEL_4:def 24</a><br/>

<b>given </b><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:73_5_1_2"/><i><font color="Green" title="E55">A46</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">m</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">m</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c12">mm</font> being   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:73_5_1_2"/><i><font color="Green" title="E56">A47</font></i>: 
( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mm</font> &amp; <font color="Maroon" title="c12">mm</font> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> )
 <b>by </b><i><a class="txt" href="armstrng.html#E6:73_5_1"><i><font color="Green" title="E51">A42</font></i></a>, <a class="txt" href="armstrng.html#E12:73_5_1"><i><font color="Green" title="E54">A45</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">e</font>, <font color="Maroon" title="c14">f</font> being   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E7:73_5_1_2"/><i><font color="Green" title="E57">A48</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">m</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">e</font>,<font color="Maroon" title="c14">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">e</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c14">f</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73_5_1_2"><i><font color="Green" title="E55">A46</font></i></a></i>;<br/>
<a NAME="E8:73_5_1_2"/><i><font color="Green" title="E58">A49</font></i>: 
( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">e</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E7:73_5_1_2"><i><font color="Green" title="E57">A48</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E9:73_5_1_2"/><b>then </b>
<font color="Maroon" title="c14">f</font> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="armstrng.html#E5:73_5_1_2"><i><font color="Green" title="E56">A47</font></i></a>, <a class="txt" href="armstrng.html#E7:73_5_1_2"><i><font color="Green" title="E57">A48</font></i></a>, <a class="ref" href="armstrng.html#T14" target="_self" title="ARMSTRNG:th.14">Th14</a></i>;<br/>
<a NAME="E10:73_5_1_2"/><b>then </b>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">MA</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E6:73_5_1"><i><font color="Green" title="E51">A42</font></i></a>, <a class="txt" href="armstrng.html#E3:73_5_1_2"><i><font color="Green" title="E55">A46</font></i></a>, <a class="txt" href="armstrng.html#E8:73_5_1_2"><i><font color="Green" title="E58">A49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:73_5_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E11:73_5_1"><i><font color="Green" title="E53">A44</font></i></a>, <a class="txt" href="armstrng.html#E3:73_5_1_2"><i><font color="Green" title="E55">A46</font></i></a>, <a class="ref" href="waybel_4.html#D24" title="WAYBEL_4:def.24">WAYBEL_4:def 24</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:73_5_1"/><b>then </b><i><font color="Green" title="E55">A50</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="armstrng.html#K1" title="ARMSTRNG:func.1">Maximal_in</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D1" target="_self" title="ARMSTRNG:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">y</font> being   <a href="armstrng.html#NM3" title="ARMSTRNG:NM.3">Dependency</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E16:73_5_1"/><i><font color="Green" title="E56">A51</font></i>: 
( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c11">y</font> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> )
 <b>by </b><i><a class="txt" href="armstrng.html#E6:73_5_1"><i><font color="Green" title="E51">A42</font></i></a>, <a class="txt" href="armstrng.html#E12:73_5_1"><i><font color="Green" title="E54">A45</font></i></a></i>;<br/><a NAME="E17:73_5_1"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:73_5_1"><i><font color="Green" title="E51">A42</font></i></a>, <a class="txt" href="armstrng.html#E12:73_5_1"><i><font color="Green" title="E54">A45</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">a'</font>, <font color="Maroon" title="c13">b'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E19:73_5_1"/><i><font color="Green" title="E57">A52</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T10" target="_self" title="ARMSTRNG:th.10">Th10</a></i>;<br/><b>thus </b><a NAME="E20:73_5_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:73"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="armstrng.html#E8:73"><i><font color="Green" title="E47">A38</font></i></a>, <a class="txt" href="armstrng.html#E3:73_5_1"><i><font color="Green" title="E49">A40</font></i></a>, <a class="txt" href="armstrng.html#E14:73_5_1"><i><font color="Green" title="E55">A50</font></i></a>, <a class="txt" href="armstrng.html#E16:73_5_1"><i><font color="Green" title="E56">A51</font></i></a>, <a class="txt" href="armstrng.html#E19:73_5_1"><i><font color="Green" title="E57">A52</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:73_5"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">a1</font>, <font color="Maroon" title="c10">b1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:73_5"/><i><font color="Green" title="E48">A53</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:73_5"/><i><font color="Green" title="E49">A54</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:73"><i><font color="Green" title="E45">A3</font></i></a></i>;<br/><b>thus </b><a NAME="E6:73_5"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:73"><i><font color="Green" title="E47">A38</font></i></a>, <a class="txt" href="armstrng.html#E4:73_5"><i><font color="Green" title="E48">A53</font></i></a>, <a class="txt" href="armstrng.html#E5:73_5"><i><font color="Green" title="E49">A54</font></i></a>, <a class="ref" href="armstrng.html#D13" target="_self" title="ARMSTRNG:def.13">Def13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:73"/>
<font color="Maroon" title="c5">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div>
