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

<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="armstrng.html#NM2" title="ARMSTRNG:NM.2">Dependency-set</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">K</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:92"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> or <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon" title="c4">BB</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c5">M</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b1">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> </span>}</span> ;<br/>
<a NAME="E2:92"/><i><font color="Green" title="E54">A2</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b1">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E1:92"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="armstrng.html#T32" target="_self" title="ARMSTRNG:th.32">Th32</a></i>;<br/>
<a NAME="E3:92"/><i><font color="Green" title="E55">A3</font></i>: 
 <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<b>set </b><font color="Maroon" title="c6">ss</font> =  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:92_1_1"/><i><font color="Green" title="E56">A4</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="armstrng.html#E1:92_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_1"><b>suppose </b></a><a NAME="E1:92_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:92_1_1_1_1"/><b>then </b><i><font color="Green" title="E57">A5</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:92_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:92_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b1">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:92_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">K</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:92"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/><b>hence </b><a NAME="E6:92_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:92"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="armstrng.html#E2:92_1_1_1_1"><i><font color="Green" title="E57">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1_2"><b>suppose </b></a><a NAME="E1:92_1_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:92_1_1_1_2"/><i><font color="Green" title="E57">A6</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font>
 <b>and </b><br/><a NAME="E4:92_1_1_1_2"/><i><font color="Green" title="E58">A7</font></i>: 
not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font>
 ;<br/><a NAME="E5:92_1_1_1_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c8">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b1">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E4:92_1_1_1_2"><i><font color="Green" title="E58">A7</font></i></a></i>;<br/><a NAME="E6:92_1_1_1_2"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c8">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b1">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:92_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">B</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:92"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/><b>hence </b><a NAME="E8:92_1_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:92_1_1_1_2"><i><font color="Green" title="E57">A6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:92_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">a</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:92_1"/><i><font color="Green" title="E56">A8</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <b>and </b><br/><a NAME="E5:92_1"/><i><font color="Green" title="E57">A9</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="armstrng.html#T33" target="_self" title="ARMSTRNG:th.33">Th33</a></i>;<br/><a NAME="E6:92_1"/><i><font color="Green" title="E58">A10</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b1">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E2:92"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="armstrng.html#E5:92_1"><i><font color="Green" title="E57">A9</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1_2"/>
( <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b1">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="armstrng.html#E6:92_1"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_1"><b>suppose </b></a><a NAME="E1:92_1_2_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:92_1_2_1"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:92_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E4:92_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:92_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E4:92_1"><i><font color="Green" title="E56">A8</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_2"><b>suppose </b></a><a NAME="E1:92_1_2_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b1">A</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">c</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:92_1_2_2"/><i><font color="Green" title="E59">A11</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c8">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c10">c</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E4:92_1_2_2"/><i><font color="Green" title="E60">A12</font></i>: 
not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 ;<br/><a NAME="E5:92_1_2_2"/><i><font color="Green" title="E61">A13</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:92_1_2_2"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E6:92_1_2_2"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E4:92_1_2_2"><i><font color="Green" title="E60">A12</font></i></a></i>;<br/><b>hence </b><a NAME="E7:92_1_2_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E4:92_1"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="armstrng.html#E5:92_1_2_2"><i><font color="Green" title="E61">A13</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:92"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : not <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div>
