<?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/>



<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:111_1"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>
 <b>and </b><br/><a NAME="E2:111_1"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">(F2)</a>
 <b>and </b><br/><a NAME="E3:111_1"/><i><font color="Green" title="E62">A3</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>
 <b>and </b><br/><a NAME="E4:111_1"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>
 ;<br/><b>thus </b><a NAME="E5:111_1"/>
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:111_1"><i><font color="Green" title="E60">A1</font></i></a></i>;<br/><b>thus </b><a NAME="E6:111_1"/>
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V13" title="ARMSTRNG:attr.13">(DC5)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font>, <font color="Maroon" title="c6">D</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#D30" target="_self" title="ARMSTRNG:def.30">ARMSTRNG:def 30</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:111_1_1"/><i><font color="Green" title="E64">A5</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">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">F</font>
 <b>and </b><br/><a NAME="E2:111_1_1"/><i><font color="Green" title="E65">A6</font></i>: 
<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="c4">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c6">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">F</font>
 ;<br/>

<a NAME="E3:111_1_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c5">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">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E1:111_1"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">Def12</a></i>;<br/>
<a NAME="E4:111_1_1"/><b>then </b>
<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="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">C</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="c2">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:111_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="armstrng.html#E1:111_1_1"><i><font color="Green" title="E64">A5</font></i></a>, <a class="ref" href="armstrng.html#D14" target="_self" title="ARMSTRNG:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E5:111_1_1"/>
<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="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c6">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">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:111_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="armstrng.html#E2:111_1_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="ref" href="armstrng.html#T20" target="_self" title="ARMSTRNG:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E7:111_1"/>
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V14" title="ARMSTRNG:attr.14">(DC6)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</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#D31" target="_self" title="ARMSTRNG:def.31">ARMSTRNG:def 31</a><br/>

<b>assume </b><a NAME="E1:111_1_2"/><i><font color="Green" title="E64">A7</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">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">F</font>
 ;<br/>

<a NAME="E2:111_1_2"/><i><font color="Green" title="E65">A8</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V7" title="ARMSTRNG:attr.7">(DC3)</a>
 
<b>by </b><i><a class="txt" href="armstrng.html#E1:111_1"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="armstrng.html#E3:111_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="armstrng.html#T24" target="_self" title="ARMSTRNG:th.24">Th24</a></i>;<br/>
<a NAME="E3:111_1_2"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:111_1_2"/><b>then </b>
<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="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c3">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="c2">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:111_1_2"><i><font color="Green" title="E65">A8</font></i></a>, <a class="ref" href="armstrng.html#D16" target="_self" title="ARMSTRNG:def.16">Def16</a></i>;<br/>
<b>hence </b><a NAME="E5:111_1_2"/>
<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="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c4">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">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:111_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="armstrng.html#E1:111_1_2"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="armstrng.html#T20" target="_self" title="ARMSTRNG:th.20">Th20</a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:111"/><i><font color="Green" title="E60">A9</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>
 <b>and </b><br/><a NAME="E3:111"/><i><font color="Green" title="E61">A10</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V13" title="ARMSTRNG:attr.13">(DC5)</a>
 <b>and </b><br/><a NAME="E4:111"/><i><font color="Green" title="E62">A11</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V14" title="ARMSTRNG:attr.14">(DC6)</a>
 ;<br/>

<b>thus </b><a NAME="E5:111"/>
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:111"><i><font color="Green" title="E60">A9</font></i></a></i>;<br/>

<div><i><font color="Green" title="E63">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">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><b>that </b><br/><a NAME="E1:111_2"/><i><font color="Green" title="E64">A13</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">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">F</font>
 <b>and </b><br/><a NAME="E2:111_2"/><i><font color="Green" title="E65">A14</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">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">F</font>
 ;<br/><a NAME="E3:111_2"/>
<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="c4">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A</font></span>)</span>,<font color="Maroon" title="c5">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">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:111"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="armstrng.html#E2:111_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="armstrng.html#D31" target="_self" title="ARMSTRNG:def.31">Def31</a></i>;<br/><a NAME="E4:111_2"/><b>then </b>
<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="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A</font></span>)</span>,<font color="Maroon" title="c5">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">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:111"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="armstrng.html#E1:111_2"><i><font color="Green" title="E64">A13</font></i></a>, <a class="ref" href="armstrng.html#D30" target="_self" title="ARMSTRNG:def.30">Def30</a></i>;<br/><b>hence </b><a NAME="E5:111_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">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">F</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:111"/>
<font color="Maroon" title="c2">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="E8:111"/>
<font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <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>; <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><b>that </b><br/><a NAME="E1:111_3"/><i><font color="Green" title="E64">A15</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">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">F</font>
 <b>and </b><br/><a NAME="E2:111_3"/><i><font color="Green" title="E65">A16</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">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>
 ;<br/>

<a NAME="E3:111_3"/><i><font color="Green" title="E66">A17</font></i>: 
( <font color="Maroon" title="c3">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="c6">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:111_3"><i><font color="Green" title="E65">A16</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/>
<a NAME="E4:111_3"/><b>then </b>
<font color="Maroon" title="c5">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E5:111_3"/><b>then </b><i><font color="Green" title="E67">A18</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="c4">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">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:111"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="armstrng.html#E1:111_3"><i><font color="Green" title="E64">A15</font></i></a>, <a class="ref" href="armstrng.html#D31" target="_self" title="ARMSTRNG:def.31">Def31</a></i>;<br/>
<a NAME="E6:111_3"/><i><font color="Green" title="E68">A19</font></i>: 
<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="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="c2">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:111"><i><font color="Green" title="E60">A9</font></i></a>, <a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">Def12</a></i>;<br/>
<a NAME="E7:111_3"/>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">B'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="armstrng.html#E3:111_3"><i><font color="Green" title="E66">A17</font></i></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E8:111_3"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">B</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="c2">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:111"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="armstrng.html#E6:111_3"><i><font color="Green" title="E68">A19</font></i></a>, <a class="ref" href="armstrng.html#D31" target="_self" title="ARMSTRNG:def.31">Def31</a></i>;<br/>
<b>hence </b><a NAME="E9:111_3"/>
<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="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:111"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="armstrng.html#E5:111_3"><i><font color="Green" title="E67">A18</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <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>; <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><b>that </b><br/><a NAME="E9:111"/><i><font color="Green" title="E64">A20</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">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">F</font>
 <b>and </b><br/><a NAME="E10:111"/><i><font color="Green" title="E65">A21</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="c2">F</font>
 ;<br/>

<a NAME="E11:111"/>
<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="c4">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">B'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">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="c2">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:111"><i><font color="Green" title="E60">A9</font></i></a>, <a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">Def12</a></i>;<br/>
<a NAME="E12:111"/><b>then </b>
<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="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">B'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">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="c2">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E3:111"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="armstrng.html#E9:111"><i><font color="Green" title="E64">A20</font></i></a>, <a class="ref" href="armstrng.html#D30" target="_self" title="ARMSTRNG:def.30">Def30</a></i>;<br/>
<b>hence </b><a NAME="E13:111"/>
<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="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">A'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">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="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:111"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="armstrng.html#E10:111"><i><font color="Green" title="E65">A21</font></i></a>, <a class="ref" href="armstrng.html#D30" target="_self" title="ARMSTRNG:def.30">Def30</a></i>;<br/>


</div>
