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<div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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><br/> for <font color="Olive" title="b2">K</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/> ( <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</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> iff <font color="Olive" title="b2">K</font> <a href="armstrng.html#R6" title="ARMSTRNG:pred.6">is_p_i_w_ncv_of</a> <font color="Olive" title="b1">F</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</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/> ( <font color="Olive" title="b1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</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> iff <font color="Olive" title="b1">K</font> <a href="armstrng.html#R6" title="ARMSTRNG:pred.6">is_p_i_w_ncv_of</a> <font color="Maroon" title="c2">F</font> )</span><br/><b>let </b><font color="Maroon" title="c3">K</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</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> iff <font color="Maroon" title="c3">K</font> <a href="armstrng.html#R6" title="ARMSTRNG:pred.6">is_p_i_w_ncv_of</a> <font color="Maroon" title="c2">F</font> )</span><br/>





<b>set </b><font color="Maroon" title="c4">dcF</font> =  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>;<br/>
<b>set </b><font color="Maroon" title="c5">ck</font> =  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</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>;<br/>
<b>set </b><font color="Maroon" title="c6">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/>
<a NAME="E1:115"/>
<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="E2:115"/><b>then </b><i><font color="Green" title="E62">A1</font></i>: 
 <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> 
 
<b>by </b><i><a class="ref" href="armstrng.html#T40" target="_self" title="ARMSTRNG:th.40">Th40</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">K</font> <a href="armstrng.html#R6" title="ARMSTRNG:pred.6">is_p_i_w_ncv_of</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c3">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</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>
<div class="add"><b>assume </b><a NAME="E1:115_1"/>
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">K</font> <a href="armstrng.html#R6" title="ARMSTRNG:pred.6">is_p_i_w_ncv_of</a> <font color="Maroon" title="c2">F</font></span><br/><b>then consider </b><font color="Maroon" title="c7">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:115_1"/><i><font color="Green" title="E63">A2</font></i>: 
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 <b>and </b><br/><a NAME="E4:115_1"/><i><font color="Green" title="E64">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c1">X</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#K8" title="ARMSTRNG:func.8">Maximal_wrt</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>
 ;<br/><a NAME="E5:115_1"/><i><font color="Green" title="E65">A4</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c1">X</font>, <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E4:115_1"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/><a NAME="E6:115_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E4:115_1"><i><font color="Green" title="E64">A3</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:115_1"/><i><font color="Green" title="E66">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c1">X</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="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>
 <b>and </b><br/><a NAME="E9:115_1"/><i><font color="Green" title="E67">A6</font></i>: 
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>
 <b>by </b><i><a class="txt" href="#E2:115"><i><font color="Green" title="E62">A1</font></i></a></i>;<br/><a NAME="E10:115_1"/><i><font color="Green" title="E68">A7</font></i>: 
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> )
 <b>by </b><i><a class="txt" href="#E8:115_1"><i><font color="Green" title="E66">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>thus </b><a NAME="E11:115_1"/>
<font color="Maroon" title="c3">K</font> <a href="armstrng.html#R6" title="ARMSTRNG:pred.6">is_p_i_w_ncv_of</a> <font color="Maroon" title="c2">F</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D33" target="_self" title="ARMSTRNG:def.33">ARMSTRNG:def 33</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">k</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">K</font> holds <br/> ex <font color="Olive" title="b2">a</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/>( <font color="Olive" title="b1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Olive" title="b2">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"><b>let </b><font color="Maroon" title="c10">z</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">z</font> &amp; <font color="Maroon" title="c10">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c10">z</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/><b>assume </b><b>that </b><br/><a NAME="E1:115_1_1_1"/><i><font color="Green" title="E69">A8</font></i>: 
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">z</font>
 <b>and </b><br/><a NAME="E2:115_1_1_1"/><i><font color="Green" title="E70">A9</font></i>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E3:115_1_1_1"/><i><font color="Green" title="E71">A10</font></i>: 
not <font color="Maroon" title="c10">z</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E4:115_1_1_1"/>
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="c10">z</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="c10">z</font>
 <b>by </b><i><a class="txt" href="#E3:115_1_1_1"><i><font color="Green" title="E71">A10</font></i></a></i>;<br/><a NAME="E5:115_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">z</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> 
 ;<br/><a NAME="E6:115_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">z</font>
 <b>by </b><i><a class="txt" href="#E3:115_1"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="#E9:115_1"><i><font color="Green" title="E67">A6</font></i></a>, <a class="txt" href="#E10:115_1"><i><font color="Green" title="E68">A7</font></i></a>, <a class="txt" href="#E1:115_1_1_1"><i><font color="Green" title="E69">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E7:115_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:115_1_1_1"><i><font color="Green" title="E70">A9</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c10">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">k</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">K</font> implies  ex <font color="Olive" title="b1">a</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/>( <font color="Maroon" title="c10">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Olive" title="b1">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/>

<b>assume </b><a NAME="E2:115_1_1"/><i><font color="Green" title="E69">A11</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a</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/>( <font color="Maroon" title="c10">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Olive" title="b1">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/>

<a NAME="E3:115_1_1"/><b>then </b>
<font color="Maroon" title="c10">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="#E3:115_1"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">k</font> = <font color="Maroon" title="c10">k</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="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>assume </b><a NAME="E5:115_1_1"/><i><font color="Green" title="E70">A12</font></i>: 
for <font color="Olive" title="b1">a</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 <font color="Maroon" title="c10">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font> or <font color="Olive" title="b1">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>set </b><font color="Maroon" title="c12">S</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</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="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<font color="Olive" title="b1">P</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>}</span> ;<br/>
<a NAME="E6:115_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</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="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<font color="Olive" title="b1">P</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>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</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="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<font color="Olive" title="b1">P</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>}</span>  or <font color="Maroon" title="c13">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> )</span><br/>

<b>assume </b><a NAME="E1:115_1_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</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="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<font color="Olive" title="b1">P</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>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">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></span><br/>

<a NAME="E2:115_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">P</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/>( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<font color="Olive" title="b1">P</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> )
 
;<br/>
<b>hence </b><a NAME="E3:115_1_1_2"/>
<font color="Maroon" title="c13">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">S</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</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="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<font color="Olive" title="b1">P</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>}</span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E8:115_1_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c11">k</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>
 
<b>by </b><i><a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">Def12</a></i>;<br/>
<a NAME="E9:115_1_1"/><b>then </b>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c14">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:115_1_1"/><i><font color="Green" title="E71">A13</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font>
 <b>and </b><br/><a NAME="E12:115_1_1"/><i><font color="Green" title="E72">A14</font></i>: 
for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font> &amp; <font color="Maroon" title="c14">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">m</font>
 <b>by </b><i><a class="ref" href="finset_1.html#T18" title="FINSET_1:th.18">FINSET_1:18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">P</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E14:115_1_1"/><i><font color="Green" title="E73">A15</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P</font>
 <b>and </b><br/><a NAME="E15:115_1_1"/><i><font color="Green" title="E74">A16</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c15">P</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>
 <b>by </b><i><a class="txt" href="#E11:115_1_1"><i><font color="Green" title="E71">A13</font></i></a></i>;<br/>
<a NAME="E16:115_1_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c11">k</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>
 
<b>by </b><i><a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">Def12</a></i>;<br/>
<a NAME="E17:115_1_1"/><b>then </b><i><font color="Green" title="E75">A17</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="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E15:115_1_1"><i><font color="Green" title="E74">A16</font></i></a>, <a class="ref" href="armstrng.html#D14" target="_self" title="ARMSTRNG:def.14">Def14</a></i>;<br/>
<a NAME="E18:115_1_1"/><i><font color="Green" title="E76">A18</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_1_1_3_1"/>
( <font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:115_1_1_3_1_1"/>
<font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><b>hence </b><a NAME="E2:115_1_1_3_1_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E17:115_1_1"><i><font color="Green" title="E75">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:115_1_1_3_1_2"/>
<font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><a NAME="E2:115_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</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>
 <b>by </b><i><a class="txt" href="#E5:115_1_1"><i><font color="Green" title="E70">A12</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:115_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</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>
 <b>by </b><i><a class="ref" href="armstrng.html#T61" target="_self" title="ARMSTRNG:th.61">Th61</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c17">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><br/><a NAME="E5:115_1_1_3_1_2"/><i><font color="Green" title="E77">A19</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c17">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>
 <b>and </b><br/><a NAME="E6:115_1_1_3_1_2"/><i><font color="Green" title="E78">A20</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font>
 <b>and </b><br/><a NAME="E7:115_1_1_3_1_2"/><i><font color="Green" title="E79">A21</font></i>: 
not <font color="Maroon" title="c17">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font>
 <b>by </b><i><a class="ref" href="armstrng.html#T57" target="_self" title="ARMSTRNG:th.57">Th57</a></i>;<br/><a NAME="E8:115_1_1_3_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="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">Def12</a></i>;<br/><a NAME="E9:115_1_1_3_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="c16">x</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font></span>)</span></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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E5:115_1_1_3_1_2"><i><font color="Green" title="E77">A19</font></i></a>, <a class="ref" href="armstrng.html#D14" target="_self" title="ARMSTRNG:def.14">Def14</a></i>;<br/><a NAME="E10:115_1_1_3_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="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font></span>)</span></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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E6:115_1_1_3_1_2"><i><font color="Green" title="E78">A20</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E11:115_1_1_3_1_2"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font></span>)</span></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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E17:115_1_1"><i><font color="Green" title="E75">A17</font></i></a>, <a class="ref" href="armstrng.html#T20" target="_self" title="ARMSTRNG:th.20">Th20</a></i>;<br/><a NAME="E12:115_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E80">A22</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">S</font>
 ;<br/><a NAME="E13:115_1_1_3_1_2"/>
<font color="Maroon" title="c15">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</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="E14:115_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c15">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E12:115_1_1"><i><font color="Green" title="E72">A14</font></i></a>, <a class="txt" href="#E14:115_1_1"><i><font color="Green" title="E73">A15</font></i></a>, <a class="txt" href="#E12:115_1_1_3_1_2"><i><font color="Green" title="E80">A22</font></i></a>, <a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></i>;<br/><b>hence </b><a NAME="E15:115_1_1_3_1_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">k</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E7:115_1_1_3_1_2"><i><font color="Green" title="E79">A21</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E19:115_1_1"/>
<font color="Maroon" title="c11">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="txt" href="#E2:115_1_1"><i><font color="Green" title="E69">A11</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E20:115_1_1"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></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="c11">k</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 
<b>by </b><i><a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E21:115_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:115_1"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="#E5:115_1"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="#E2:115_1_1"><i><font color="Green" title="E69">A11</font></i></a>, <a class="txt" href="#E18:115_1_1"><i><font color="Green" title="E76">A18</font></i></a>, <a class="ref" href="armstrng.html#T29" target="_self" title="ARMSTRNG:th.29">Th29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E4:115"/><i><font color="Green" title="E63">A23</font></i>: 
<font color="Maroon" title="c3">K</font> <a href="armstrng.html#R6" title="ARMSTRNG:pred.6">is_p_i_w_ncv_of</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</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/>

<b>set </b><font color="Maroon" title="c7">S</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</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="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Olive" title="b1">P</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>}</span> ;<br/>
<a NAME="E5:115"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</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="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Olive" title="b1">P</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>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</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="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Olive" title="b1">P</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>}</span>  or <font color="Maroon" title="c8">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> )</span><br/>

<b>assume </b><a NAME="E1:115_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</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="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Olive" title="b1">P</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>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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></span><br/>

<a NAME="E2:115_2"/><b>then </b>
 ex <font color="Olive" title="b1">P</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/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Olive" title="b1">P</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> )
 
;<br/>
<b>hence </b><a NAME="E3:115_2"/>
<font color="Maroon" title="c8">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">S</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</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="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Olive" title="b1">P</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>}</span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E7:115"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c3">K</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>
 
<b>by </b><i><a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">Def12</a></i>;<br/>
<a NAME="E8:115"/><b>then </b>
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c9">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:115"/><i><font color="Green" title="E64">A24</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font>
 <b>and </b><br/><a NAME="E11:115"/><i><font color="Green" title="E65">A25</font></i>: 
for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> &amp; <font color="Maroon" title="c9">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="ref" href="finset_1.html#T18" title="FINSET_1:th.18">FINSET_1:18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">P</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E13:115"/><i><font color="Green" title="E66">A26</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P</font>
 <b>and </b><br/><a NAME="E14:115"/><i><font color="Green" title="E67">A27</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c10">P</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>
 <b>by </b><i><a class="txt" href="#E10:115"><i><font color="Green" title="E64">A24</font></i></a></i>;<br/>
<a NAME="E15:115"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c3">K</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>
 
<b>by </b><i><a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">Def12</a></i>;<br/>
<a NAME="E16:115"/><b>then </b><i><font color="Green" title="E68">A28</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="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">K</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E14:115"><i><font color="Green" title="E67">A27</font></i></a>, <a class="ref" href="armstrng.html#D14" target="_self" title="ARMSTRNG:def.14">Def14</a></i>;<br/>
<a NAME="E17:115"/><i><font color="Green" title="E69">A29</font></i>: 
<font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</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="E18:115"/><i><font color="Green" title="E70">A30</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_3_1"/>
( <font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_3_1_1"><b>suppose </b></a><a NAME="E1:115_3_1_1"/>
<font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">K</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><b>hence </b><a NAME="E2:115_3_1_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E16:115"><i><font color="Green" title="E68">A28</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_3_1_2"><b>suppose </b></a><a NAME="E1:115_3_1_2"/>
<font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">K</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><a NAME="E2:115_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</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>
 <b>by </b><i><a class="txt" href="#E4:115"><i><font color="Green" title="E63">A23</font></i></a>, <a class="txt" href="#E17:115"><i><font color="Green" title="E69">A29</font></i></a>, <a class="ref" href="armstrng.html#D33" target="_self" title="ARMSTRNG:def.33">Def33</a></i>;<br/><a NAME="E3:115_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</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>
 <b>by </b><i><a class="ref" href="armstrng.html#T61" target="_self" title="ARMSTRNG:th.61">Th61</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">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><br/><a NAME="E5:115_3_1_2"/><i><font color="Green" title="E71">A31</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">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>
 <b>and </b><br/><a NAME="E6:115_3_1_2"/><i><font color="Green" title="E72">A32</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font>
 <b>and </b><br/><a NAME="E7:115_3_1_2"/><i><font color="Green" title="E73">A33</font></i>: 
not <font color="Maroon" title="c12">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font>
 <b>by </b><i><a class="ref" href="armstrng.html#T57" target="_self" title="ARMSTRNG:th.57">Th57</a></i>;<br/><a NAME="E8:115_3_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">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">Def12</a></i>;<br/><a NAME="E9:115_3_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="c11">x</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font></span>)</span></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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E5:115_3_1_2"><i><font color="Green" title="E71">A31</font></i></a>, <a class="ref" href="armstrng.html#D14" target="_self" title="ARMSTRNG:def.14">Def14</a></i>;<br/><a NAME="E10:115_3_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">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font></span>)</span></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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E6:115_3_1_2"><i><font color="Green" title="E72">A32</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E11:115_3_1_2"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font></span>)</span></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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E16:115"><i><font color="Green" title="E68">A28</font></i></a>, <a class="ref" href="armstrng.html#T20" target="_self" title="ARMSTRNG:th.20">Th20</a></i>;<br/><a NAME="E12:115_3_1_2"/><b>then </b><i><font color="Green" title="E74">A34</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font>
 ;<br/><a NAME="E13:115_3_1_2"/>
<font color="Maroon" title="c10">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</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="E14:115_3_1_2"/><b>then </b>
<font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:115"><i><font color="Green" title="E65">A25</font></i></a>, <a class="txt" href="#E13:115"><i><font color="Green" title="E66">A26</font></i></a>, <a class="txt" href="#E12:115_3_1_2"><i><font color="Green" title="E74">A34</font></i></a>, <a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></i>;<br/><b>hence </b><a NAME="E15:115_3_1_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</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>  <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E7:115_3_1_2"><i><font color="Green" title="E73">A33</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x'</font>, <font color="Maroon" title="c12">y'</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">x'</font>,<font color="Maroon" title="c12">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> &amp; ( <font color="Maroon" title="c3">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x'</font> or <font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y'</font> ) implies  not <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></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="c11">x'</font>,<font color="Maroon" title="c12">y'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:115_4"/><i><font color="Green" title="E71">A35</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">x'</font>,<font color="Maroon" title="c12">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>
 <b>and </b><br/><a NAME="E2:115_4"/><i><font color="Green" title="E72">A36</font></i>: 
( <font color="Maroon" title="c3">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x'</font> or <font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y'</font> )
 <b>and </b><br/><a NAME="E3:115_4"/><i><font color="Green" title="E73">A37</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></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="c11">x'</font>,<font color="Maroon" title="c12">y'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E4:115_4"/><i><font color="Green" title="E74">A38</font></i>: 
( <font color="Maroon" title="c11">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">y'</font> )
 <b>by </b><i><a class="txt" href="#E3:115_4"><i><font color="Green" title="E73">A37</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><a NAME="E5:115_4"/><b>then </b><i><font color="Green" title="E75">A39</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E6:115_4"/>
<font color="Maroon" title="c11">x'</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">K</font>
 <b>by </b><i><a class="txt" href="#E2:115_4"><i><font color="Green" title="E72">A36</font></i></a>, <a class="txt" href="#E4:115_4"><i><font color="Green" title="E74">A38</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">a</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:115_4"/><i><font color="Green" title="E76">A40</font></i>: 
<font color="Maroon" title="c11">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">a</font>
 <b>and </b><br/><a NAME="E9:115_4"/><i><font color="Green" title="E77">A41</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E10:115_4"/><i><font color="Green" title="E78">A42</font></i>: 
not <font color="Maroon" title="c13">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>
 <b>by </b><i><a class="txt" href="#E4:115"><i><font color="Green" title="E63">A23</font></i></a>, <a class="ref" href="armstrng.html#D33" target="_self" title="ARMSTRNG:def.33">Def33</a></i>;<br/><a NAME="E11:115_4"/><i><font color="Green" title="E79">A43</font></i>: 
not <font color="Maroon" title="c13">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>
 <b>by </b><i><a class="txt" href="#E10:115_4"><i><font color="Green" title="E78">A42</font></i></a>, <a class="ref" href="armstrng.html#T61" target="_self" title="ARMSTRNG:th.61">Th61</a></i>;<br/><a NAME="E12:115_4"/>
not <font color="Maroon" title="c12">y'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">a</font>
 <b>by </b><i><a class="txt" href="#E5:115_4"><i><font color="Green" title="E75">A39</font></i></a>, <a class="txt" href="#E9:115_4"><i><font color="Green" title="E77">A41</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E13:115_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:115_4"><i><font color="Green" title="E71">A35</font></i></a>, <a class="txt" href="#E8:115_4"><i><font color="Green" title="E76">A40</font></i></a>, <a class="txt" href="#E11:115_4"><i><font color="Green" title="E79">A43</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E20:115"/><b>then </b>
<font color="Maroon" title="c3">K</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c1">X</font>, <a href="armstrng.html#K12" title="ARMSTRNG:func.12">Dependency-closure</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E18:115"><i><font color="Green" title="E70">A30</font></i></a>, <a class="ref" href="armstrng.html#T29" target="_self" title="ARMSTRNG:th.29">Th29</a></i>;<br/>
<a NAME="E21:115"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</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#K8" title="ARMSTRNG:func.8">Maximal_wrt</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>
 
<b>by </b><i><a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/>
<b>hence </b><a NAME="E22:115"/>
<font color="Maroon" title="c3">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
