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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="armstrng.html#NM5">Full-family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="armstrng.html#T35" target="_self">Th35</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E2:83"/><i><font color="Green">E48</font></i>: 
 <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">c<sub>5</sub></font> is <a href="armstrng.html#V8" target="_self">(M1)</a>
 
<b>by </b><i><a class="ref" href="armstrng.html#T30" target="_self">Th30</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></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="E3:83"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E3:83"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:83"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E8:83"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></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">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:83"><i><font color="Green">E48</font></i></a>, <a class="ref" href="armstrng.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E9:83"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E7:83"><i><font color="Green">E50</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E10:83"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E8:83"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:83"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></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">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E13:83"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>
<a NAME="E14:83"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="armstrng.html#E12:83"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E15:83"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E3:83"><i><font color="Green">E49</font></i></a>, <a class="txt" href="armstrng.html#E9:83"><i><font color="Green">E52</font></i></a>, <a class="txt" href="armstrng.html#E13:83"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E16:83"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E9:83"><i><font color="Green">E52</font></i></a>, <a class="txt" href="armstrng.html#E14:83"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E17:83"/>
<font color="Maroon">c<sub>12</sub></font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E8:83"><i><font color="Green">E51</font></i></a>, <a class="txt" href="armstrng.html#E12:83"><i><font color="Green">E53</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self">Def18</a></i>;<br/>
<b>hence </b><a NAME="E18:83"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="armstrng.html#K10" target="_self">deps_encl_by</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E16:83"><i><font color="Green">E56</font></i></a></i>;<br/>


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