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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">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>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</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>set </b><font color="Maroon" title="c5">DOX</font> =  <a href="armstrng.html#K7" title="ARMSTRNG:func.7">Dependencies-Order</a> <font color="Maroon" title="c1">X</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_1"/>
<font color="Maroon" title="c3">x</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>
 ;<br/><a NAME="E2:70_1"/><b>then </b><i><font color="Green" title="E38">A1</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <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="c2">F</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/><a NAME="E3:70_1"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c2">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>thus </b><a NAME="E4:70_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</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">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:70_1"><i><font color="Green" title="E38">A1</font></i></a></i>;<br/><b>given </b><font color="Maroon" title="c6">x'</font>, <font color="Maroon" title="c7">y'</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><a NAME="E6:70_1"/><i><font color="Green" title="E40">A3</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">x'</font>,<font color="Maroon" title="c7">y'</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">F</font> &amp; ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x'</font> or <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y'</font> ) &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">x'</font>,<font color="Maroon" title="c7">y'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> )
 ;<br/><a NAME="E7:70_1"/><i><font color="Green" title="E41">A4</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">x'</font>,<font color="Maroon" title="c7">y'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:70_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E8:70_1"/>
<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="c3">x</font>,<font color="Maroon" title="c4">y</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="c6">x'</font>,<font color="Maroon" title="c7">y'</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#E6:70_1"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E9:70_1"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E3:70_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="armstrng.html#E6:70_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="armstrng.html#E7:70_1"><i><font color="Green" title="E41">A4</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>assume </b><a NAME="E2:70"/><i><font color="Green" title="E38">A5</font></i>: 
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</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">F</font> &amp; ( for <font color="Olive" title="b1">x'</font>, <font color="Olive" title="b2">y'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  holds <br/>(  not <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">x'</font>,<font color="Olive" title="b2">y'</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">F</font> or ( not <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x'</font> &amp; not <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y'</font> ) or  not <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">x'</font>,<font color="Olive" title="b2">y'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> ) ) )
 ;<br/>

<a NAME="E3:70"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="waybel_4.html#R2" title="WAYBEL_4:pred.2">is_maximal_wrt</a> <font color="Maroon" title="c2">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="70_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:70_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</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">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:70"><i><font color="Green" title="E38">A5</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="c6">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:70_2"/><i><font color="Green" title="E39">A6</font></i>: 
( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <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="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>,<font color="Maroon" title="c6">z</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="c7">x'</font>, <font color="Maroon" title="c8">y'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:70_2"/><i><font color="Green" title="E40">A7</font></i>: 
( <font color="Maroon" title="c6">z</font> <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="c7">x'</font>,<font color="Maroon" title="c8">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c8">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:70_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x'</font> = <font color="Maroon" title="c7">x'</font>, <font color="Maroon" title="c10">y'</font> = <font color="Maroon" title="c8">y'</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="armstrng.html#E5:70_2"><i><font color="Green" title="E40">A7</font></i></a></i>;<br/>
<a NAME="E7:70_2"/><i><font color="Green" title="E41">A8</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">x'</font> or <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y'</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E3:70_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="armstrng.html#E5:70_2"><i><font color="Green" title="E40">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">e</font>, <font color="Maroon" title="c12">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="E9:70_2"/><i><font color="Green" title="E42">A9</font></i>: 
( <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="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>,<font color="Maroon" title="c6">z</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="c11">e</font>,<font color="Maroon" title="c12">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">e</font> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <font color="Maroon" title="c12">f</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:70_2"><i><font color="Green" title="E39">A6</font></i></a></i>;<br/>
<a NAME="E10:70_2"/>
( <font color="Maroon" title="c11">e</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="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> &amp; <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E9:70_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E11:70_2"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E2:70"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="armstrng.html#E3:70_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="armstrng.html#E5:70_2"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="armstrng.html#E7:70_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="armstrng.html#E9:70_2"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:70"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <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="c2">F</font>
 
<b>by </b><i><a class="ref" href="armstrng.html#D1" target="_self" title="ARMSTRNG:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:70"/>
<font color="Maroon" title="c3">x</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/>


</div>
