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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">B</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>set </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1"/>
(  not <font color="Maroon" title="c2">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a> or <font color="Maroon" title="c2">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><b>suppose </b></a><a NAME="E1:82_1_1"/><i><font color="Green" title="E46">A1</font></i>: 
not <font color="Maroon" title="c2">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:82_1_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font> )
 ;<br/><b>assume </b><a NAME="E2:82_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:82_1_1_1"/><i><font color="Green" title="E47">A2</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E5:82_1_1_1"/>
for <font color="Olive" title="b1">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Maroon" title="c7">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">g</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:82_1_1"><i><font color="Green" title="E46">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E6:82_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:82_1_1_1"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:82_1_1"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="armstrng.html#K4" title="ARMSTRNG:func.4">Dependencies</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E4:82_1_1"/><b>then </b>
( <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a> &amp; <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">(F2)</a> &amp; <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a> &amp; <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a> )
 <b>by </b><i><a class="ref" href="armstrng.html#T21" target="_self" title="ARMSTRNG:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E5:82_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a>
 <b>by </b><i><a class="ref" href="armstrng.html#D15" target="_self" title="ARMSTRNG:def.15">Def15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><b>suppose </b></a><a NAME="E1:82_1_2"/>
<font color="Maroon" title="c2">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c2">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/><div><i><font color="Green" title="E46">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c8">c</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">B</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:82_1_2_1"/><i><font color="Green" title="E47">A4</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E2:82_1_2_1"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font>
 ;<br/><b>consider </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">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="E4:82_1_2_1"/><i><font color="Green" title="E49">A6</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:82_1_2_1"/><i><font color="Green" title="E50">A7</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> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c9">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="c10">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="armstrng.html#E1:82_1_2_1"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/><a NAME="E6:82_1_2_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E4:82_1_2_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E7:82_1_2_1"/>
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:82_1_2_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="armstrng.html#E5:82_1_2_1"><i><font color="Green" title="E50">A7</font></i></a></i>;<br/></div><b>end;</b></div><b>thus </b><a NAME="E4:82_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D15" target="_self" title="ARMSTRNG:def.15">ARMSTRNG:def 15</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D12" target="_self" title="ARMSTRNG:def.12">ARMSTRNG:def 12</a><br/>

<a NAME="E1:82_1_2_2"/>
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> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c6">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="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font>
 
;<br/>
<b>hence </b><a NAME="E2:82_1_2_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c6">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:82_1_2_3"/><i><font color="Green" title="E47">A8</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E2:82_1_2_3"/><i><font color="Green" title="E48">A9</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:82_1_2_3_1"/><i><font color="Green" title="E49">A10</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">x</font> )
 ;<br/><a NAME="E2:82_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:82_1_2"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="armstrng.html#E1:82_1_2_3"><i><font color="Green" title="E47">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E3:82_1_2_3_1"/>
<font color="Maroon" title="c8">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:82_1_2"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="armstrng.html#E2:82_1_2_3"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="armstrng.html#E1:82_1_2_3_1"><i><font color="Green" title="E49">A10</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:82_1_2_3"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:82_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">(F2)</a>
 <b>by </b><i><a class="ref" href="armstrng.html#T20" target="_self" title="ARMSTRNG:th.20">Th20</a></i>;<br/><b>thus </b><a NAME="E7:82_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">a'</font>, <font color="Maroon" title="c9">b'</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D13" target="_self" title="ARMSTRNG:def.13">ARMSTRNG:def 13</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:82_1_2_4"/><i><font color="Green" title="E47">A11</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E2:82_1_2_4"/><i><font color="Green" title="E48">A12</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 ;<br/>

<a NAME="E3:82_1_2_4"/><i><font color="Green" title="E49">A13</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:82_1_2_4"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:82_1_2_4_1"/><i><font color="Green" title="E50">A14</font></i>: 
( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c8">a'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font> )
 ;<br/><a NAME="E2:82_1_2_4_1"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:82_1_2_4"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:82_1_2_4_1"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:82_1_2"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="armstrng.html#E1:82_1_2_4"><i><font color="Green" title="E47">A11</font></i></a>, <a class="txt" href="armstrng.html#E1:82_1_2_4_1"><i><font color="Green" title="E50">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E4:82_1_2_4_1"/>
<font color="Maroon" title="c9">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:82_1_2_4"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:82_1_2_4"/>
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E8:82_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font> is <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">a'</font>, <font color="Maroon" title="c9">b'</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D14" target="_self" title="ARMSTRNG:def.14">ARMSTRNG:def 14</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:82_1_2_5"/><i><font color="Green" title="E47">A15</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E2:82_1_2_5"/><i><font color="Green" title="E48">A16</font></i>: 
<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:82_1_2_5_1"/><i><font color="Green" title="E49">A17</font></i>: 
( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c6">a</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">a'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font> )
 ;<br/><a NAME="E2:82_1_2_5_1"/><b>then </b>
( <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c8">a'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a></i>;<br/><a NAME="E3:82_1_2_5_1"/><b>then </b>
( <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:82_1_2"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="armstrng.html#E1:82_1_2_5"><i><font color="Green" title="E47">A15</font></i></a>, <a class="txt" href="armstrng.html#E2:82_1_2_5"><i><font color="Green" title="E48">A16</font></i></a>, <a class="txt" href="armstrng.html#E1:82_1_2_5_1"><i><font color="Green" title="E49">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E4:82_1_2_5_1"/>
<font color="Maroon" title="c7">b</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">b'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:82_1_2_5"/>
<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="c6">a</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">a'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">b'</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c2">B</font>
 ;<br/>


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

</div>
