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<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="armstrng.html#NM2">Dependency-set</a> of <font color="Maroon">X</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:110_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V3" target="_self">(F1)</a>
 <b>and </b><br/><a NAME="E2:110_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> is <a href="relat_2.html#V8">(F2)</a>
 <b>and </b><br/><a NAME="E3:110_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V4" target="_self">(F3)</a>
 <b>and </b><br/><a NAME="E4:110_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V5" target="_self">(F4)</a>
 ;<br/><b>thus </b><a NAME="E5:110_1"/>
<font color="Maroon">F</font> is <a href="relat_2.html#V8">(F2)</a>
 <b>by </b><i><a class="ref" href="armstrng.html#D1" target="_self">Def1</a></i>;<br/><b>thus </b><a NAME="E6:110_1"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V7" target="_self">(DC3)</a>
 <b>by </b><i>, , <a class="txt" href="armstrng.html#E1:110_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>thus </b><a NAME="E7:110_1"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V12" target="_self">(DC4)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D29" target="_self">ARMSTRNG:def 29</a><br/>

<b>assume </b><a NAME="E1:110_1_1"/>
( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">C</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:110_1_1"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="armstrng.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>hence </b><a NAME="E3:110_1_1"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A</font>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:110"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">F</font> is <a href="relat_2.html#V8">(F2)</a>
 <b>and </b><br/><a NAME="E3:110"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V7" target="_self">(DC3)</a>
 <b>and </b><br/><a NAME="E4:110"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> is <a href="armstrng.html#V12" target="_self">(DC4)</a>
 ;<br/>

<b>thus </b><a NAME="E5:110"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V3" target="_self">(F1)</a>
 <b>by </b><i>, <a class="ref" href="armstrng.html#D1" target="_self">Def1</a>, </i>;<br/>

<b>thus </b><a NAME="E6:110"/>
<font color="Maroon">F</font> is <a href="relat_2.html#V8">(F2)</a>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E7:110"/>
<font color="Maroon">F</font> is <a href="armstrng.html#V4" target="_self">(F3)</a>
 <b>by </b><i>, <a class="ref" href="armstrng.html#D1" target="_self">Def1</a>, </i>;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">A'</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B'</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D14" target="_self">ARMSTRNG:def 14</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E8:110"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E9:110"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A'</font>,<font color="Maroon">B'</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E10:110"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E11:110"/><b>then </b>
( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font></span>)</span>,<font color="Maroon">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font></span>)</span>,<font color="Maroon">A'</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="armstrng.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E12:110"/><b>then </b>
( <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font></span>)</span>,<font color="Maroon">B</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font></span>)</span>,<font color="Maroon">B'</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="armstrng.html#D2" target="_self">Def2</a>, <a class="ref" href="armstrng.html#D3" target="_self">Def3</a>, </i>;<br/>
<b>hence </b><a NAME="E13:110"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font></span>)</span></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


</div>
