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

<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>assume </b><a NAME="E1:84"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="armstrng.html#V1" target="_self">(B1)</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="finsub_1.html#V2">(B2)</a> )
 ;<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="E3:84"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E1:84"><i><font color="Green">E49</font></i></a>, <a class="ref" href="armstrng.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="armstrng.html#E1:84"><i><font color="Green">E49</font></i></a>, <a class="ref" href="armstrng.html#D4" target="_self">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:84_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</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="ref" href="armstrng.html#T36" target="_self">Th36</a></i>;<br/><b>hence </b><a NAME="E2:84_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>8</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> )
 ;<br/><b>assume </b><a NAME="E3:84_1"/>
<font color="Maroon">c<sub>8</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>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</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="E5:84_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E6:84_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">c<sub>9</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="ref" href="armstrng.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E7:84_1"/>
<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>9</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#E6:84_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E8:84_1"/><b>then </b>
<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>9</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>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</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="E10:84_1"/><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>9</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>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E11:84_1"/><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>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><a NAME="E12:84_1"/><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>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E10:84_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> : <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ;<br/><a NAME="E13:84_1"/><i><font color="Green">E56</font></i>: 
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T24">FINSET_1:24</a></i>;<br/><a NAME="E14:84_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> : <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</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="E1:84_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> : <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E3:84_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/>
<a NAME="E4:84_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E5:84_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:84_1_1"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:84_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> : <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="armstrng.html#E13:84_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E16:84_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> : <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E3:84"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E17:84_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> : <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</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="E1:84_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> : <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E3:84_1_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:84_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:84_1_2"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> : <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E14:84_1"><i><font color="Green">E57</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>14</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>14</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E20:84_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</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="E1:84_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:84_1_3_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E3:84_1_3_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E4:84_1_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E11:84_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="armstrng.html#E12:84_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="armstrng.html#E3:84_1_3_1"><i><font color="Green">E63</font></i></a></i>;<br/><b>hence </b><a NAME="E5:84_1_3_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:84_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="armstrng.html#E3:84_1_3_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:84_1_3"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E16:84_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:84_1_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E16:84_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:84_1_4_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E2:84_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E3:84_1_4_1"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/><b>hence </b><a NAME="E4:84_1_4_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E16:84_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/></div><b>end;</b></div><a NAME="E3:84_1_4"/><b>then </b><i><font color="Green">E63</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>16</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>
 ;<br/><a NAME="E4:84_1_4"/>
<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>16</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>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>by </b><i><a class="txt" href="armstrng.html#E20:84_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E5:84_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E6:84_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="armstrng.html#E1:84_1_4"><i><font color="Green">E62</font></i></a>, <a class="txt" href="armstrng.html#E3:84_1_4"><i><font color="Green">E63</font></i></a>, <a class="ref" href="armstrng.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E22:84_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:84"><i><font color="Green">E49</font></i></a>, <a class="txt" href="armstrng.html#E3:84"><i><font color="Green">E50</font></i></a>, <a class="txt" href="armstrng.html#E5:84_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="armstrng.html#E15:84_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="armstrng.html#E17:84_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="armstrng.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:84"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</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="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="armstrng.html#NM5">Full-family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E7:84"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K9" target="_self">saturated-subsets</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E8:84"/><i><font color="Green">E52</font></i>: 
(  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">c<sub>8</sub></font> is <a href="armstrng.html#V8" target="_self">(M1)</a> &amp;  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">c<sub>8</sub></font> is <a href="armstrng.html#V10" target="_self">(M3)</a> )
 
<b>by </b><i><a class="ref" href="armstrng.html#T30" target="_self">Th30</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:84_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="armstrng.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>8</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="armstrng.html#NM3">Dependency</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</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="E5:84_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <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>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T9" target="_self">Th9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:84_2_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E2:84_2_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</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="E4:84_2_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E5:84_2_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E7:84"><i><font color="Green">E51</font></i></a>, <a class="txt" href="armstrng.html#E1:84_2_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="armstrng.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E6:84_2_1_1"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:84_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="armstrng.html#E5:84_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self">Def18</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</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="E8:84_2_1_1"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</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>8</sub></font>
 <b>and </b><br/><a NAME="E9:84_2_1_1"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="armstrng.html#R3" target="_self">&gt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="armstrng.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E10:84_2_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:84_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="armstrng.html#E9:84_2_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E11:84_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:84_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:84_2_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:84"><i><font color="Green">E52</font></i></a>, <a class="txt" href="armstrng.html#E4:84_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="armstrng.html#E6:84_2_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="armstrng.html#E8:84_2_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="armstrng.html#D21" target="_self">Def21</a></i>;<br/><a NAME="E13:84_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:84_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="armstrng.html#E9:84_2_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E14:84_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:84_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="armstrng.html#E12:84_2_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:84_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <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#E5:84_2_1"><i><font color="Green">E53</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:84_2"/>
<font color="Maroon">c<sub>10</sub></font> <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>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</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="E4:84_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <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>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E5:84_2"/><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>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</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="E7:84_2"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</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>11</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E8:84_2"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</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>8</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:84"><i><font color="Green">E52</font></i></a>, <a class="ref" href="armstrng.html#D19" target="_self">Def19</a></i>;<br/><a NAME="E9:84_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E7:84_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E10:84_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="armstrng.html#R4" target="_self">^|^</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:84_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E11:84_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E7:84"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E12:84_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:84_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="armstrng.html#E9:84_2"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E13:84_2"/><b>then </b>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</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>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>by </b><i><a class="txt" href="armstrng.html#E9:84_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E14:84_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:84_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="armstrng.html#E8:84_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="armstrng.html#D13" target="_self">Def13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:84"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</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="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
