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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <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   <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Olive" title="b1">F</font></span>)</span></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">  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">dcF</font> =  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">B</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">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="Olive" title="b2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Olive" title="b3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> </span> } </span> ;<br/><span class="kw">let </span><font color="Maroon" title="c5">A</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"> ( ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span>)</span> ) &amp; ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span>)</span> implies <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c2">F</font> ) )</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="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span>)</span> implies <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c2">F</font> )</span>
<div class="add"><a NAME="E1:115_1_1"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#D24" target="_self" title="ARMSTRNG:def.24">Def25</a></span>;<br/><span class="kw">assume </span><a NAME="E2:115_1_1"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</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"> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/><a NAME="E3:115_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A3</font></span>: 
<font color="Maroon" title="c5">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T55" target="_self" title="ARMSTRNG:th.55">Th57</a></span>;<br/><a NAME="E4:115_1_1"/>
 ex <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> st <br/>( <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> &amp; <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> &amp;  not <font color="Olive" title="b2">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_1"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="armstrng.html#T55" target="_self" title="ARMSTRNG:th.55">Th57</a></span>;<br/><span class="kw">hence </span><a NAME="E5:115_1_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1_1"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E3:115_1_1"><span class="lab"><font color="Green" title="E64">A3</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">assume </span><a NAME="E2:115_1"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span>)</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="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c2">F</font></span><br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:115_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="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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">and </span><br/><a NAME="E5:115_1"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">and </span><br/><a NAME="E6:115_1"/><span class="lab"><font color="Green" title="E65">A7</font></span>: 
 not <font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T55" target="_self" title="ARMSTRNG:th.55">Th57</a></span>;<br/><a NAME="E7:115_1"/><span class="lab"><font color="Green" title="E66">A8</font></span>: 
<font color="Maroon" title="c5">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="armstrng.html#T55" target="_self" title="ARMSTRNG:th.55">Th57</a></span>;<br/><span class="kw">assume </span><a NAME="E8:115_1"/>
 not <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</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"> contradiction</span><br/><a NAME="E9:115_1"/><span class="kw">then </span>
 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>  st <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> &amp; <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> holds <br/><font color="Olive" title="b2">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:115_1"><span class="lab"><font color="Green" title="E66">A8</font></span></a></span>;<br/><a NAME="E10:115_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A9</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">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="Olive" title="b2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Olive" title="b3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:115_1"><span class="lab"><font color="Green" title="E66">A8</font></span></a></span>;<br/><a NAME="E11:115_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">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="Olive" title="b2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Olive" title="b3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K11" title="ARMSTRNG:func.11">enclosure_of</a> <font color="Maroon" title="c2">F</font>
 ;<br/><a NAME="E12:115_1"/><span class="kw">then </span>
 <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">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="Olive" title="b2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Olive" title="b3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="armstrng.html#T38" target="_self" title="ARMSTRNG:th.38">Th40</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E14:115_1"/><span class="lab"><font color="Green" title="E68">A10</font></span>: 
<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#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="c8">a</font>,<font color="Maroon" title="c9">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E15:115_1"/><span class="lab"><font color="Green" title="E69">A11</font></span>: 
 for <font color="Olive" title="b1">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">c</font> where <font color="Olive" title="b2">c</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">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="Olive" title="b3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">c</font> holds <br/><font color="Olive" title="b4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">c</font> </span> } </span>  &amp; <font color="Maroon" title="c8">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c9">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a></span>;<br/><a NAME="E16:115_1"/><span class="lab"><font color="Green" title="E70">A12</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:115_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E17:115_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:115_1"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E18:115_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:115_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E6:115_1"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E15:115_1"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="txt" href="#E16:115_1"><span class="lab"><font color="Green" title="E70">A12</font></span></a>, <a class="txt" href="#E10:115_1"><span class="lab"><font color="Green" title="E67">A9</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">hence </span><a NAME="E2:115"/>
 <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K14" title="ARMSTRNG:func.14">charact_set</a> <span class="p1">(<span class="default"><a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</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/>


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