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

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="armstrng.html#NM2">Dependency-set</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:60_2_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="armstrng.html#V7" target="_self">(DC3)</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="relat_2.html#V8">(F2)</a> )
 ;<br/>

<b>thus </b><a NAME="E2:60_2_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="armstrng.html#V3" target="_self">(F1)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D12" target="_self">ARMSTRNG:def 12</a><br/>

<b>thus </b><a NAME="E1:60_2_1_1"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:60_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="armstrng.html#D16" target="_self">Def16</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D13" target="_self">ARMSTRNG:def 13</a><br/>

<b>assume </b><a NAME="E3:60_2_1"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>assume </b><a NAME="E4:60_2_1"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="armstrng.html#R3" target="_self">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 ;<br/>

<a NAME="E5:60_2_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="armstrng.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E6:60_2_1"/><b>then </b>
( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E1:60_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="armstrng.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E7:60_2_1"/><b>then </b>
( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E1:60_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="armstrng.html#E3:60_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="armstrng.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E8:60_2_1"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:60_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="armstrng.html#T20" target="_self">Th20</a></i>;<br/>


</div>
