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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <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#NM5">Full-family</a> of <font color="Maroon">X</font>;<br/>



<b>set </b><font color="Maroon">ss</font> =  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">F</font>;<br/>
<a NAME="E1:80"/><i><font color="Green">E33</font></i>: 
 <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font> is <a href="armstrng.html#V8" target="_self">(M1)</a>
 
<b>by </b><i/>;<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="E3:80"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A'</font>,<font color="Maroon">B'</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="armstrng.html#R3" target="_self">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span>,<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>
 <b>and </b><br/><a NAME="E4:80"/><i><font color="Green">E35</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>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:80"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E5:80"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A'</font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">B'</font>,<font color="Maroon">F</font>
 
<b>by </b><i>, <a class="txt" href="armstrng.html#E4:80"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E6:80"/>
 <a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:80"/><b>then </b>
 <a href="subset_1.html#K2">[#]</a> <font color="Maroon">X</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/>
<a NAME="E8:80"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:80"/>
 <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">F</font> is <a href="armstrng.html#V1" target="_self">(B1)</a>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E10:80"/>
 <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">F</font> is <a href="finsub_1.html#V2">(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">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:80_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:80_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">F</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="armstrng.html#E10:80_1"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">B1</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">A1</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:80_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 <b>and </b><br/><a NAME="E6:80_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">A1</font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">B1</font>,<font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E10:80_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="armstrng.html#E6:80_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E7:80_1"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">B1</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="txt" href="armstrng.html#E4:80"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">B2</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">A2</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:80_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font>
 <b>and </b><br/><a NAME="E10:80_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">A2</font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">B2</font>,<font color="Maroon">F</font>
 <b>by </b><i>, <a class="txt" href="armstrng.html#E6:80_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E11:80_1"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">A2</font>,<font color="Maroon">B2</font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="txt" href="armstrng.html#E4:80"><i><font color="Green">E35</font></i></a></i>;<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="E13:80_1"/><i><font color="Green">E75</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"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b'</font></span>)</span></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E14:80_1"/><i><font color="Green">E79</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>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E9:80_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="armstrng.html#E5:80"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E15:80_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">A'</font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">B'</font>,<font color="Maroon">F</font>
 
<b>by </b><i>, <a class="txt" href="armstrng.html#E4:80"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E16:80_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> <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="E17:80_1"/>
<font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> <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#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E18:80_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:80_1"/><i><font color="Green">E83</font></i>: 
 <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">F</font> is <a href="armstrng.html#V10" target="_self">(M3)</a>
 
<b>by </b><i/>;<br/>
<a NAME="E20:80_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="armstrng.html#T2" target="_self">Th2</a>, , <a class="ref" href="armstrng.html#D1" target="_self">Def1</a>, <a class="txt" href="armstrng.html#E2:80_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E21:80_1"/>
<font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> <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="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E22:80_1"/><b>then </b>
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E23:80_1"/><b>then </b>
<font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, , , <a class="txt" href="armstrng.html#E2:80_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E24:80_1"/><b>then </b>
<font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="armstrng.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E25:80_1"/><b>then </b>
<font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</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/>
<b>hence </b><a NAME="E26:80_1"/>
<font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


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


</div>
