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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="armstrng.html#NV4" title="ARMSTRNG:NV.4">(F2)</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>  iff ( <font color="Olive" title="b1">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="armstrng.html#V13" title="ARMSTRNG:attr.13">(DC5)</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="armstrng.html#V14" title="ARMSTRNG:attr.14">(DC6)</a>  ) )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#NV4" title="ARMSTRNG:NV.4">(F2)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>  iff ( <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V13" title="ARMSTRNG:attr.13">(DC5)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V14" title="ARMSTRNG:attr.14">(DC6)</a>  ) )</span><br/>



<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V13" title="ARMSTRNG:attr.13">(DC5)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V14" title="ARMSTRNG:attr.14">(DC6)</a>  implies ( <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#NV4" title="ARMSTRNG:NV.4">(F2)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>  ) )</span>
<div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:110_1"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a> 
 <span class="kw">and </span><br/><a NAME="E2:110_1"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#NV4" title="ARMSTRNG:NV.4">(F2)</a> 
 <span class="kw">and </span><br/><a NAME="E3:110_1"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a> 
 <span class="kw">and </span><br/><a NAME="E4:110_1"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V13" title="ARMSTRNG:attr.13">(DC5)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V14" title="ARMSTRNG:attr.14">(DC6)</a>  )</span><br/><span class="kw">thus </span><a NAME="E5:110_1"/>
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V13" title="ARMSTRNG:attr.13">(DC5)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V14" title="ARMSTRNG:attr.14">(DC6)</a>  )</span><br/><span class="kw">thus </span><a NAME="E6:110_1"/>
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V13" title="ARMSTRNG:attr.13">(DC5)</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V14" title="ARMSTRNG:attr.14">(DC6)</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="armstrng.html#D29" target="_self" title="ARMSTRNG:def.29">ARMSTRNG:def 29</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> implies <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> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:110_1_1"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:110_1_1"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E3:110_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="armstrng.html#D11" target="_self" title="ARMSTRNG:def.11">Def12</a></span>;<br/>
<a NAME="E4:110_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E1:110_1_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="armstrng.html#D13" target="_self" title="ARMSTRNG:def.13">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:110_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E2:110_1_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="ref" href="armstrng.html#T18" target="_self" title="ARMSTRNG:th.18">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E7:110_1"/>
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V14" title="ARMSTRNG:attr.14">(DC6)</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="armstrng.html#D30" target="_self" title="ARMSTRNG:def.30">ARMSTRNG:def 30</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:110_1_2"/><span class="lab"><font color="Green" title="E63">A7</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:110_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E3:110_1_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E3:110_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="ref" href="armstrng.html#D15" target="_self" title="ARMSTRNG:def.15">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:110_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E1:110_1_2"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="ref" href="armstrng.html#T18" target="_self" title="ARMSTRNG:th.18">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:110"/><span class="lab"><font color="Green" title="E59">A8</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a> 
 <span class="kw">and </span><br/><a NAME="E3:110"/><span class="lab"><font color="Green" title="E60">A9</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V13" title="ARMSTRNG:attr.13">(DC5)</a> 
 <span class="kw">and </span><br/><a NAME="E4:110"/><span class="lab"><font color="Green" title="E61">A10</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V14" title="ARMSTRNG:attr.14">(DC6)</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#NV4" title="ARMSTRNG:NV.4">(F2)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E5:110"/>
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E59">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#NV4" title="ARMSTRNG:NV.4">(F2)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>  )</span><br/>

<div><span class="lab"><font color="Green" title="E62">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:110_2"/><span class="lab"><font color="Green" title="E63">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:110_2"/><span class="lab"><font color="Green" title="E64">A13</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><a NAME="E3:110_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E2:110_2"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="ref" href="armstrng.html#D30" target="_self" title="ARMSTRNG:def.30">Def31</a></span>;<br/><a NAME="E4:110_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E1:110_2"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="ref" href="armstrng.html#D29" target="_self" title="ARMSTRNG:def.29">Def30</a></span>;<br/><span class="kw">hence </span><a NAME="E5:110_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:110"/>
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#NV4" title="ARMSTRNG:NV.4">(F2)</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T18" target="_self" title="ARMSTRNG:th.18">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E8:110"/>
<font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> is  <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">A9</font>, <font color="Maroon" title="c6">B9</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">ARMSTRNG:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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">A9</font>,<font color="Maroon" title="c6">B9</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> implies <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">B9</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> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:110_3"/><span class="lab"><font color="Green" title="E63">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:110_3"/><span class="lab"><font color="Green" title="E64">A15</font></span>: 
<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">A9</font>,<font color="Maroon" title="c6">B9</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">B9</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></span><br/>

<a NAME="E3:110_3"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_3"><span class="lab"><font color="Green" title="E64">A15</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/>
<a NAME="E4:110_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">A9</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">A9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<a NAME="E5:110_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A16</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A9</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E1:110_3"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="ref" href="armstrng.html#D30" target="_self" title="ARMSTRNG:def.30">Def31</a></span>;<br/>
<a NAME="E6:110_3"/>
<font color="Maroon" title="c6">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_3"><span class="lab"><font color="Green" title="E64">A15</font></span></a>, <a class="ref" href="armstrng.html#T13" target="_self" title="ARMSTRNG:th.13">Th15</a></span>;<br/>
<a NAME="E7:110_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A17</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B9</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">B9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<a NAME="E8:110_3"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">B9</font>,<font color="Maroon" title="c6">B9</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="ref" href="armstrng.html#D11" target="_self" title="ARMSTRNG:def.11">Def12</a></span>;<br/>
<a NAME="E9:110_3"/><span class="kw">then </span>
<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">B9</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E7:110_3"><span class="lab"><font color="Green" title="E66">A17</font></span></a>, <a class="ref" href="armstrng.html#D30" target="_self" title="ARMSTRNG:def.30">Def31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:110_3"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">B9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E5:110_3"><span class="lab"><font color="Green" title="E65">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">A9</font>, <font color="Maroon" title="c6">B9</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="armstrng.html#D13" target="_self" title="ARMSTRNG:def.13">ARMSTRNG:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">B9</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> implies <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">A9</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">B9</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> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:110"/><span class="lab"><font color="Green" title="E63">A18</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E10:110"/><span class="lab"><font color="Green" title="E64">A19</font></span>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A9</font>,<font color="Maroon" title="c6">B9</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">A9</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">B9</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></span><br/>

<a NAME="E11:110"/>
<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">B9</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">B9</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="ref" href="armstrng.html#D11" target="_self" title="ARMSTRNG:def.11">Def12</a></span>;<br/>
<a NAME="E12:110"/><span class="kw">then </span>
<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">B9</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">B9</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E9:110"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="ref" href="armstrng.html#D29" target="_self" title="ARMSTRNG:def.29">Def30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:110"/>
<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">A9</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">B9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E10:110"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="ref" href="armstrng.html#D29" target="_self" title="ARMSTRNG:def.29">Def30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
