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

<b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a>  <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/><b>let </b><font color="Maroon">K</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:75"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><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> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> or <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:75"/><i><font color="Green">E34</font></i>: 
 <a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
;<br/>
<b>set </b><font color="Maroon">M</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span> ;<br/>
<a NAME="E3:75"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:75"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:75"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>now </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>;<br/><b>assume </b><a NAME="E1:75_1"/>
not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:75_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<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> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">a</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> </span>}</span> 
 ;<br/><a NAME="E3:75_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">a</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:75_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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2"><b>now </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>;<br/><b>assume </b><a NAME="E1:75_2"/><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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_2_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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or <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> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">a</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="armstrng.html#E10:75"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_2_1_1"><b>suppose </b></a><a NAME="E1:75_2_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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:75_2_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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span> or ( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_2_1_2"><b>suppose </b></a><a NAME="E1:75_2_1_2"/>
<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> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">a</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:75_2_1_2"/><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#R1">=</a> <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> &amp; not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 ;<br/><a NAME="E4:75_2_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:75_2_1_2"/>
( <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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span> or ( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:75"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:75_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="armstrng.html#E10:75"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_3_1_1"><b>suppose </b></a><a NAME="E1:75_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:75_3_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:75_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_3_1_2"><b>suppose </b></a><a NAME="E1:75_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:75_3_1_2"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> &amp; not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> )
 ;<br/><a NAME="E3:75_3_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> &amp;  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 ;<br/><b>hence </b><a NAME="E4:75_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span>  as   <a href="armstrng.html#NM2">Dependency-set</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E10:75"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">M</font> is <a href="armstrng.html#V8" target="_self">(M1)</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_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>; <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D19" target="_self">ARMSTRNG:def 19</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_4_1"/>
( <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> or  not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_4_1_1"><b>suppose </b></a><a NAME="E1:75_4_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">A'</font> = <font color="Maroon">K</font>;<br/><b>take </b><font color="Maroon">B'</font> =  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</font>;<br/><b>thus </b><a NAME="E2:75_4_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="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">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E3:75_4_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">M</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_4_1_2"><b>suppose </b></a><a NAME="E1:75_4_1_2"/><i><font color="Green">E75</font></i>: 
not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">A'</font> = <font color="Maroon">A</font>;<br/><b>take </b><font color="Maroon">B'</font> = <font color="Maroon">A</font>;<br/><b>thus </b><a NAME="E2:75_4_1_2"/>
<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">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 ;<br/><b>thus </b><a NAME="E3:75_4_1_2"/>
<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">M</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:75"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">M</font> is <a href="armstrng.html#V9" target="_self">(M2)</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_5"><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#D20" target="_self">ARMSTRNG:def 20</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:75_5"/><i><font color="Green">E80</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">M</font>
 <b>and </b><br/><a NAME="E2:75_5"/><i><font color="Green">E81</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">M</font>
 <b>and </b><br/><a NAME="E3:75_5"/><i><font color="Green">E82</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="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>
 ;<br/>

<a NAME="E4:75_5"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="armstrng.html#T2" target="_self">Th2</a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_5_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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span> or ( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_5_1_1"><b>suppose </b></a><a NAME="E1:75_5_1_1"/><i><font color="Green">E84</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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:75_5_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_5_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_5_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_5_1_1_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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span> or ( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_5_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:75_5_1_1_1_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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:75_5_1_1_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_5_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:75_5_1_1_1_1_2"/>
( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:75_5_1_1_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_5_1_2"><b>suppose </b></a><a NAME="E1:75_5_1_2"/><i><font color="Green">E85</font></i>: 
( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:75_5_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_5_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_5_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_5_1_2_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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span> or ( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:75_5_1_2_1_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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:75_5_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E3:75_5_1_2_1_1_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E4:75_5_1_2_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:75_5_1_2_1_1_2"/>
( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:75_5_1_2_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div>
<a NAME="E12:75"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">M</font> is <a href="armstrng.html#V10" target="_self">(M3)</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_6"><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#D21" target="_self">ARMSTRNG:def 21</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:75_6"/><i><font color="Green">E110</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">M</font>
 <b>and </b><br/><a NAME="E2:75_6"/><i><font color="Green">E111</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">M</font>
 <b>and </b><br/><a NAME="E3:75_6"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_6_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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span> or ( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_6_1_1"><b>suppose </b></a><a NAME="E1:75_6_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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:75_6_1_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:75_6_1_1"/>
<font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_6_1_2"><b>suppose </b></a><a NAME="E1:75_6_1_2"/><i><font color="Green">E113</font></i>: 
( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:75_6_1_2"/>
<font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_6_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_6_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_6_1_2_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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span> or ( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> ) )
 </span><b>by </b><i>, <a class="ref" href="armstrng.html#T2" target="_self">Th2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_6_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:75_6_1_2_1_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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:75_6_1_2_1_1_1"/>
<font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_6_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:75_6_1_2_1_1_2"/>
( not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:75_6_1_2_1_1_2"/>
<font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div>
<b>set </b><font color="Maroon">FF</font> = <span class="p1">{<span class="default"> <span class="p2"><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> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">A'</font>, <font color="Olive">B'</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <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="armstrng.html#R3" target="_self">&gt;=</a> <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> &amp; <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">M</font> ) </span>}</span> ;<br/>
<a NAME="E13:75"/>
<span class="p1">{<span class="default"> <span class="p2"><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> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">A'</font>, <font color="Olive">B'</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <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="armstrng.html#R3" target="_self">&gt;=</a> <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> &amp; <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">M</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:75_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><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> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">A'</font>, <font color="Olive">B'</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <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="armstrng.html#R3" target="_self">&gt;=</a> <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> &amp; <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">M</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:75_7"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</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>
 <b>and </b><br/><a NAME="E4:75_7"/>
 ex <font color="Olive">A'</font>, <font color="Olive">B'</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <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="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> &amp; <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">M</font> )
 ;<br/>
<a NAME="E5:75_7"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> &amp;  <a href="armstrng.html#K4" target="_self">Dependencies</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:75_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">FF</font> = <span class="p1">{<span class="default"> <span class="p2"><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> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">A'</font>, <font color="Olive">B'</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <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="armstrng.html#R3" target="_self">&gt;=</a> <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> &amp; <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">M</font> ) </span>}</span>  as   <a href="armstrng.html#NM2">Dependency-set</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E15:75"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">FF</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E10:75"><i><font color="Green">E50</font></i></a>, , , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:75_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:75_8_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</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>
 <b>and </b><br/><a NAME="E4:75_8_1"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> or <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><a NAME="E5:75_8_1"/><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E6:75_8_1"/><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:75_8_1"/><i><font color="Green">E120</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">a</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_8_1_1"/>
( <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> or  not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_8_1_1_1"><b>suppose </b></a><a NAME="E1:75_8_1_1_1"/>
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:75_8_1_1_1"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">K</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span></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>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:75_8_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 <b>by </b><i><a class="ref" href="armstrng.html#T2" target="_self">Th2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_8_1_1_2"><b>suppose </b></a><a NAME="E1:75_8_1_1_2"/><i><font color="Green">E121</font></i>: 
not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:75_8_1_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<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> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">a</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> </span>}</span> 
 ;<br/><a NAME="E3:75_8_1_1_2"/>
<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="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>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:75_8_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 <b>by </b><i><a class="ref" href="armstrng.html#T2" target="_self">Th2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:75_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 ;<br/><b>then consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:75_8"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</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>
 <b>and </b><br/><a NAME="E5:75_8"/><i><font color="Green">E124</font></i>: 
 ex <font color="Olive">A'</font>, <font color="Olive">B'</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <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="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> &amp; <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">M</font> )
 ;<br/><b>consider </b><font color="Maroon">A'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:75_8"/><i><font color="Green">E125</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="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> &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> <font color="Maroon">M</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_8_2"/>
( <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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> or <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> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">a</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_8_2_1"><b>suppose </b></a><a NAME="E1:75_8_2_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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:75_8_2_1"/><b>then </b>
<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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:75_8_2_1"/><b>then </b>
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E4:75_8_2_1"/><b>then </b>
( <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:75_8_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="armstrng.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_8_2_2"><b>suppose </b></a><a NAME="E1:75_8_2_2"/>
<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> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">a</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">a</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:75_8_2_2"/><i><font color="Green">E126</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#R1">=</a> <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>
 <b>and </b><br/><a NAME="E4:75_8_2_2"/>
not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E5:75_8_2_2"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E6:75_8_2_2"/>
( <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B'</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:75_8_2_2"/><b>then </b>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E8:75_8_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="armstrng.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:75"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,<font color="Maroon">X</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Olive">A</font>,<font color="Olive">A</font></span><a href="armstrng.html#K6" target="_self">]</a></span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
