<?xml version="1.0"?>
<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><a NAME="E1:49_1"/>
<font color="Maroon">F</font> is <a href="relat_2.html#V8">(F2)</a>
 ;<br/><a NAME="E2:49_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D16">RELAT_2:def 16</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">C</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E3:49_1"/><i><font color="Green">E34</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> &amp; <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">B</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="E4:49_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">F</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">F</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>hence </b><a NAME="E5:49_1"/>
<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>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma26</font></i></a>, , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:49"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">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="Olive">B</font>,<font color="Olive">C</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">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/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a>,<a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a><br/>

<b>assume </b><a NAME="E3:49"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">F</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">F</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E4:49"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T19">RELSET_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">x</font>, <font color="Maroon">B</font> = <font color="Maroon">y</font>, <font color="Maroon">C</font> = <font color="Maroon">z</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E6:49"/>
( <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">B</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> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:49"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma26</font></i></a></i>;<br/>


</div>
