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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <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="E1:36"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/><a NAME="E4:36_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:36"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow_9.html#E3:36_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/><a NAME="E5:36_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>1</sub></font> &amp; <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:36_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a>, <a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>;<br/><b>hence </b><a NAME="E6:36_1"/>
<font color="Maroon">c<sub>3</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="yellow_9.html#E1:36"><i><font color="Green">E14</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E3:36"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cantor_1.html#T1">CANTOR_1:1</a></i>;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:36_2"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/><a NAME="E4:36_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:36"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow_9.html#E3:36_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/><a NAME="E5:36_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:36_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><a NAME="E6:36_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>1</sub></font> &amp; <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a>, <a class="ref" href="setfam_1.html#T12">SETFAM_1:12</a></i>;<br/><b>hence </b><a NAME="E7:36_2"/>
<font color="Maroon">c<sub>3</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="yellow_9.html#E1:36"><i><font color="Green">E14</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E5:36"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cantor_1.html#T4">CANTOR_1:4</a></i>;<br/>


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