<?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>set </b><font color="Maroon">D</font> =  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font>;<br/>
<a NAME="E1:51"/>
 <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K1">nabla</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="eqrel_1.html#D1">EQREL_1:def 1</a></i>;<br/>
<a NAME="E2:51"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<b>thus </b><a NAME="E3:51"/>
 <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> is <a href="armstrng.html#V3" target="_self">(F1)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_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>; <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="E1:51_1"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font>
 ;<br/>


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

<b>thus </b><a NAME="E4:51"/>
 <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> is <a href="relat_2.html#V8">(F2)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<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="E1:51_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font></span>)</span> &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>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</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>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> )
 ;<br/>

<b>hence </b><a NAME="E2:51_2"/>
<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>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E5:51"/>
 <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> is <a href="armstrng.html#V4" target="_self">(F3)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><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">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#D13" target="_self">ARMSTRNG:def 13</a><br/>

<b>thus </b><a NAME="E1:51_3"/>
( <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>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> &amp; <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="armstrng.html#R3" target="_self">&gt;=</a> <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> implies <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>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> )
 ;<br/>


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

<b>thus </b><a NAME="E6:51"/>
 <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> is <a href="armstrng.html#V5" target="_self">(F4)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4"><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">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>thus </b><a NAME="E1:51_4"/>
( <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>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> &amp; <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>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> implies <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>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> )
 ;<br/>


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


</div>
