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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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#NM5" title="ARMSTRNG:NM.5">Full-family</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>set </b><font color="Maroon" title="c3">ss</font> =  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>;<br/>
<a NAME="E1:80"/><i><font color="Green" title="E46">A1</font></i>: 
 <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V8" title="ARMSTRNG:attr.8">(M1)</a>
 
<b>by </b><i><a class="ref" href="armstrng.html#T30" target="_self" title="ARMSTRNG:th.30">Th30</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">A'</font>, <font color="Maroon" title="c5">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="E3:80"/><i><font color="Green" title="E47">A2</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c5">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E4:80"/><i><font color="Green" title="E48">A3</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c4">A'</font>,<font color="Maroon" title="c5">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>  <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="ref" href="armstrng.html#D19" target="_self" title="ARMSTRNG:def.19">Def19</a></i>;<br/>
<a NAME="E5:80"/><i><font color="Green" title="E49">A4</font></i>: 
<font color="Maroon" title="c4">A'</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c5">B'</font>,<font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:80"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/>
<a NAME="E6:80"/>
 <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B'</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E3:80"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/>
<a NAME="E7:80"/><b>then </b>
 <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="c5">B'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E8:80"/><b>then </b>
<font color="Maroon" title="c1">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#E5:80"><i><font color="Green" title="E49">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:80"/>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V1" title="ARMSTRNG:attr.1">(B1)</a>
 <b>by </b><i><a class="ref" href="armstrng.html#D4" target="_self" title="ARMSTRNG:def.4">Def4</a></i>;<br/>

<b>thus </b><a NAME="E10:80"/>
 <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">(B2)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:80_1"/><i><font color="Green" title="E50">A5</font></i>: 
<font color="Maroon" title="c6">a</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>and </b><br/><a NAME="E2:80_1"/><i><font color="Green" title="E51">A6</font></i>: 
<font color="Maroon" title="c7">b</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>reconsider </b><font color="Maroon" title="c8">a'</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b'</font> = <font color="Maroon" title="c7">b</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="armstrng.html#E1:80_1"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="armstrng.html#E2:80_1"><i><font color="Green" title="E51">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">B1</font>, <font color="Maroon" title="c11">A1</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="E5:80_1"/><i><font color="Green" title="E52">A7</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font>
 <b>and </b><br/><a NAME="E6:80_1"/><i><font color="Green" title="E53">A8</font></i>: 
<font color="Maroon" title="c11">A1</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c10">B1</font>,<font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:80_1"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="armstrng.html#T33" target="_self" title="ARMSTRNG:th.33">Th33</a></i>;<br/>
<a NAME="E7:80_1"/><i><font color="Green" title="E54">A9</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c11">A1</font>,<font color="Maroon" title="c10">B1</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#E6:80_1"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">B2</font>, <font color="Maroon" title="c13">A2</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="E9:80_1"/><i><font color="Green" title="E55">A10</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B2</font>
 <b>and </b><br/><a NAME="E10:80_1"/><i><font color="Green" title="E56">A11</font></i>: 
<font color="Maroon" title="c13">A2</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c12">B2</font>,<font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:80_1"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="armstrng.html#T33" target="_self" title="ARMSTRNG:th.33">Th33</a></i>;<br/>
<a NAME="E11:80_1"/><i><font color="Green" title="E57">A12</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c13">A2</font>,<font color="Maroon" title="c12">B2</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#E10:80_1"><i><font color="Green" title="E56">A11</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">A'</font>, <font color="Maroon" title="c15">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="E13:80_1"/><i><font color="Green" title="E58">A13</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c14">A'</font>,<font color="Maroon" title="c15">B'</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">a'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b'</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E14:80_1"/><i><font color="Green" title="E59">A14</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c14">A'</font>,<font color="Maroon" title="c15">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>  <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#E1:80"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="armstrng.html#D19" target="_self" title="ARMSTRNG:def.19">Def19</a></i>;<br/>
<a NAME="E15:80_1"/><i><font color="Green" title="E60">A15</font></i>: 
<font color="Maroon" title="c14">A'</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a> <font color="Maroon" title="c15">B'</font>,<font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E14:80_1"><i><font color="Green" title="E59">A14</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/>
<a NAME="E16:80_1"/><i><font color="Green" title="E61">A16</font></i>: 
( <font color="Maroon" title="c14">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c6">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">B'</font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E13:80_1"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/>
<a NAME="E17:80_1"/>
<font color="Maroon" title="c6">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E18:80_1"/><b>then </b><i><font color="Green" title="E62">A17</font></i>: 
<font color="Maroon" title="c14">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B1</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E5:80_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="armstrng.html#E16:80_1"><i><font color="Green" title="E61">A16</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:80_1"/><i><font color="Green" title="E63">A18</font></i>: 
 <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <font color="Maroon" title="c2">F</font> is <a href="armstrng.html#V10" title="ARMSTRNG:attr.10">(M3)</a>
 
<b>by </b><i><a class="ref" href="armstrng.html#T30" target="_self" title="ARMSTRNG:th.30">Th30</a></i>;<br/>
<a NAME="E20:80_1"/><b>then </b><i><font color="Green" title="E64">A19</font></i>: 
<font color="Maroon" title="c15">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B1</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E7:80_1"><i><font color="Green" title="E54">A9</font></i></a>, <a class="txt" href="armstrng.html#E14:80_1"><i><font color="Green" title="E59">A14</font></i></a>, <a class="txt" href="armstrng.html#E18:80_1"><i><font color="Green" title="E62">A17</font></i></a>, <a class="ref" href="armstrng.html#D21" target="_self" title="ARMSTRNG:def.21">Def21</a></i>;<br/>
<a NAME="E21:80_1"/>
<font color="Maroon" title="c6">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E22:80_1"/><b>then </b>
<font color="Maroon" title="c14">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">B2</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E9:80_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="txt" href="armstrng.html#E16:80_1"><i><font color="Green" title="E61">A16</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E23:80_1"/><b>then </b>
<font color="Maroon" title="c15">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">B2</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E11:80_1"><i><font color="Green" title="E57">A12</font></i></a>, <a class="txt" href="armstrng.html#E14:80_1"><i><font color="Green" title="E59">A14</font></i></a>, <a class="txt" href="armstrng.html#E19:80_1"><i><font color="Green" title="E63">A18</font></i></a>, <a class="ref" href="armstrng.html#D21" target="_self" title="ARMSTRNG:def.21">Def21</a></i>;<br/>
<a NAME="E24:80_1"/><b>then </b>
<font color="Maroon" title="c15">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E5:80_1"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="armstrng.html#E9:80_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="txt" href="armstrng.html#E20:80_1"><i><font color="Green" title="E64">A19</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E25:80_1"/><b>then </b>
<font color="Maroon" title="c15">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E16:80_1"><i><font color="Green" title="E61">A16</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E26:80_1"/>
<font color="Maroon" title="c6">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">b</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#E15:80_1"><i><font color="Green" title="E60">A15</font></i></a></i>;<br/>


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


</div>
