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

<b>let </b><font color="Maroon" title="c1">D</font> be    <a href="armstrng.html#L1" title="ARMSTRNG:struct.1">DB-Rel</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="armstrng.html#V6" title="ARMSTRNG:attr.6">full_family</a></span><br/>

<b>set </b><font color="Maroon" title="c2">S</font> =  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font>;<br/>
<b>set </b><font color="Maroon" title="c3">T</font> = the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font>;<br/>
<b>set </b><font color="Maroon" title="c4">R</font> = the <a href="armstrng.html#U3" title="ARMSTRNG:sel.3">Relationship</a> of <font color="Maroon" title="c1">D</font>;<br/>
<a NAME="E1:62"/><i><font color="Green" title="E32">A1</font></i>: 
 <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="armstrng.html#V7" title="ARMSTRNG:attr.7">(DC3)</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D16" target="_self" title="ARMSTRNG:def.16">ARMSTRNG:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> implies <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> )</span><br/>

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</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="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span><br/>

<a NAME="E2:62_1"/>
<font color="Maroon" title="c5">A</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c1">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="armstrng.html#U3" title="ARMSTRNG:sel.3">Relationship</a> of <font color="Maroon" title="c1">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">ARMSTRNG:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font> implies <font color="Maroon" title="c7">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font> )</span><br/>

<b>assume </b><a NAME="E1:62_1_1"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font></span><br/>

<b>thus </b><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="txt" href="#E1:62_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>
<br/>.= 
<font color="Maroon" title="c8">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="txt" href="#E1:62_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="#E1:62_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</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>hence </b><a NAME="E3:62_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</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>
<div><i><font color="Green" title="E33">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font>, <font color="Maroon" title="c7">C</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</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="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">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>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> implies <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">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>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> )</span><br/><b>assume </b><a NAME="E1:62_2"/>
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">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>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</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="c5">A</font>,<font color="Maroon" title="c7">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>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span><br/><a NAME="E2:62_2"/><b>then </b><i><font color="Green" title="E34">A5</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c6">B</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="ref" href="armstrng.html#T12" target="_self" title="ARMSTRNG:th.12">Th12</a></i>;<br/><a NAME="E3:62_2"/>
<font color="Maroon" title="c5">A</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c1">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="armstrng.html#U3" title="ARMSTRNG:sel.3">Relationship</a> of <font color="Maroon" title="c1">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">ARMSTRNG:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font> implies <font color="Maroon" title="c8">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">C</font> )</span><br/>

<b>assume </b><a NAME="E1:62_2_1"/>
<font color="Maroon" title="c8">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">C</font></span><br/>

<a NAME="E2:62_2_1"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="#E2:62_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E3:62_2_1"/>
<font color="Maroon" title="c8">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="txt" href="#E2:62_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">Def8</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>hence </b><a NAME="E4:62_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">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>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</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>
<a NAME="E3:62"/><b>then </b><i><font color="Green" title="E34">A6</font></i>: 
 <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</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>hence </b><a NAME="E4:62"/>
 <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="armstrng.html#V3" title="ARMSTRNG:attr.3">(F1)</a>
 <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E32">A1</font></i></a></i>; <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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">(F2)</a> &amp;  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a> &amp;  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a> )</span><br/>

<b>thus </b><a NAME="E5:62"/>
 <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">(F2)</a>
 <b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="armstrng.html#T20" target="_self" title="ARMSTRNG:th.20">Th20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a> &amp;  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a> )</span><br/>

<b>thus </b><a NAME="E6:62"/>
 <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="armstrng.html#V4" title="ARMSTRNG:attr.4">(F3)</a>
 <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="#E3:62"><i><font color="Green" title="E34">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a></span><br/>

<b>thus </b><a NAME="E7:62"/>
 <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> is <a href="armstrng.html#V5" title="ARMSTRNG:attr.5">(F4)</a>
  <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="62_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font>, <font color="Maroon" title="c7">A'</font>, <font color="Maroon" title="c8">B'</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c1">D</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> <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="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c8">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>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> implies <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">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>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> )</span><br/>

<b>assume </b><a NAME="E1:62_3"/>
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font> &amp; <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c7">A'</font>,<font color="Maroon" title="c8">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>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">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>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</font></span><br/>

<a NAME="E2:62_3"/><b>then </b><i><font color="Green" title="E35">A7</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c1">D</font> &amp; <font color="Maroon" title="c7">A'</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c8">B'</font>,<font color="Maroon" title="c1">D</font> )
 
<b>by </b><i><a class="ref" href="armstrng.html#T12" target="_self" title="ARMSTRNG:th.12">Th12</a></i>;<br/>
<a NAME="E3:62_3"/>
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font>,<font color="Maroon" title="c1">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="armstrng.html#U3" title="ARMSTRNG:sel.3">Relationship</a> of <font color="Maroon" title="c1">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">ARMSTRNG:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span> implies <font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:62_3_1"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</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="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font></span>)</span></span><br/>

<font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="txt" href="#E1:62_3_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>
;<br/>
<a NAME="E3:62_3_1"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="#E2:62_3"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">Def8</a></i>;<br/>
<font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">A'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">A'</font>
<b>by </b><i><a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c7">A'</font>
<b>by </b><i><a class="txt" href="#E1:62_3_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>
;<br/>
<a NAME="E5:62_3_1"/><b>then </b><i><font color="Green" title="E38">A10</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">B'</font>
 
<b>by </b><i><a class="txt" href="#E2:62_3"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">Def8</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">B'</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T107" title="RELAT_1:th.107">RELAT_1:107</a></i>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:62_3_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="#E5:62_3_1"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="relat_1.html#T107" title="RELAT_1:th.107">RELAT_1:107</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>hence </b><a NAME="E4:62_3"/>
<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="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">A'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">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>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c1">D</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>


</div>
