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

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:5_1_1"/><i><font color="Green">E4</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> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( 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> holds <br/> <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R2">are_equipotent</a>  or <font color="Olive">b<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="ref" href="zfmisc_1.html#T136">ZFMISC_1:136</a></i>;<br/>
<a NAME="E3:5_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="classes1.html#V1" target="_self">subset-closed</a>
 
<b>by </b><i><a class="txt" href="classes1.html#E2:5_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="classes1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:5_1_1"/><b>then </b><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a> 
 
<b>by </b><i><a class="txt" href="classes1.html#E2:5_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="classes1.html#D2" target="_self">Def2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="classes1.html#R1" target="_self">is_Tarski-Class_of</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:5_1_1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E7:5_1_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="classes1.html#R1" target="_self">is_Tarski-Class_of</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="classes1.html#E2:5_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="classes1.html#E4:5_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="classes1.html#D3" target="_self">Def3</a>, <a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/>
<a NAME="E8:5_1_1"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="classes1.html#E6:5_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E9:5_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="classes1.html#E6:5_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="classes1.html#E7:5_1_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<div><i><font color="Green">E10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:5_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="classes1.html#R1" target="_self">is_Tarski-Class_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E6:5_1_1"><i><font color="Green">E6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:5_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="classes1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:5_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="classes1.html#E9:5_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>take </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E12:5_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E9:5_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="classes1.html#E10:5_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D3" target="_self">CLASSES1:def 3</a><br/>

<b>thus </b><a NAME="E13:5_1_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D1" target="_self">CLASSES1:def 1</a>,<a class="ref" href="classes1.html#D2" target="_self">CLASSES1:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:5_1_1_2"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_1_2_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:5_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="classes1.html#R1" target="_self">is_Tarski-Class_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E6:5_1_1"><i><font color="Green">E6</font></i></a></i>;<br/><a NAME="E3:5_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a> 
 <b>by </b><i><a class="ref" href="classes1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E4:5_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is <a href="classes1.html#V1" target="_self">subset-closed</a>
 <b>by </b><i><a class="ref" href="classes1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:5_1_1_2_1"/><b>then </b>
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="classes1.html#E1:5_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="classes1.html#E1:5_1_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="classes1.html#D1" target="_self">Def1</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:5_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E1:5_1_1_2"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:5_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E9:5_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E14:5_1_1"/><i><font color="Green">E13</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>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:5_1_1_3"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_1_3_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:5_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="classes1.html#R1" target="_self">is_Tarski-Class_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E6:5_1_1"><i><font color="Green">E6</font></i></a></i>;<br/><a NAME="E3:5_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a> 
 <b>by </b><i><a class="ref" href="classes1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E4:5_1_1_3_1"/><b>then </b>
( ( 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>6</sub></font> holds <br/> <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="classes1.html#E1:5_1_1_3"><i><font color="Green">E14</font></i></a>, <a class="txt" href="classes1.html#E1:5_1_1_3_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="classes1.html#D2" target="_self">Def2</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E5:5_1_1_3_1"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:5_1_1_3"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E9:5_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E15:5_1_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font>, <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R2">are_equipotent</a>  or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:5_1_1_4"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:5_1_1_4"/><i><font color="Green">E16</font></i>: 
not <font color="Maroon">c<sub>5</sub></font>, <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_1_4_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:5_1_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="classes1.html#R1" target="_self">is_Tarski-Class_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E6:5_1_1"><i><font color="Green">E6</font></i></a></i>;<br/><a NAME="E3:5_1_1_4_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a> 
 <b>by </b><i><a class="ref" href="classes1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E4:5_1_1_4_1"/><i><font color="Green">E19</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E1:5_1_1_4_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/><a NAME="E5:5_1_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E1:5_1_1_4"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:5_1_1_4_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R2">are_equipotent</a>  or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="classes1.html#E3:5_1_1_4_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="classes1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E7:5_1_1_4_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E1:5_1_1_4"><i><font color="Green">E15</font></i></a>, <a class="txt" href="classes1.html#E2:5_1_1_4"><i><font color="Green">E16</font></i></a>, <a class="txt" href="classes1.html#E4:5_1_1_4_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="card_1.html#T44">CARD_1:44</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:5_1_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E9:5_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E16:5_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="classes1.html#R1" target="_self">is_Tarski-Class_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E17:5_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a>  )
 
<b>by </b><i><a class="ref" href="classes1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E18:5_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="classes1.html#V1" target="_self">subset-closed</a>
 
<b>by </b><i><a class="ref" href="classes1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E19:5_1_1"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="classes1.html#R1" target="_self">is_Tarski-Class_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:5_1_1_5"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E11:5_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="classes1.html#E17:5_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D3" target="_self">CLASSES1:def 3</a><br/>

<b>thus </b><a NAME="E2:5_1_1_5"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D1" target="_self">CLASSES1:def 1</a>,<a class="ref" href="classes1.html#D2" target="_self">CLASSES1:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:5_1_1_5_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E2:5_1_1_5_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:5_1_1_5_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="classes1.html#E13:5_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="classes1.html#E18:5_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="classes1.html#E1:5_1_1_5_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="classes1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:5_1_1_5_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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

<b>thus </b><a NAME="E3:5_1_1_5"/>
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> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:5_1_1_5_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:5_1_1_5_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:5_1_1_5_2"/><b>then </b>
(  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="classes1.html#E14:5_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="classes1.html#E17:5_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="classes1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:5_1_1_5_2"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E4:5_1_1_5"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E5:5_1_1_5"/><i><font color="Green">E19</font></i>: 
not <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 ;<br/>

<a NAME="E6:5_1_1_5"/><i><font color="Green">E20</font></i>: 
( <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E7:5_1_1_5"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="classes1.html#E4:5_1_1_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:5_1_1_5"/>
( not <font color="Maroon">c<sub>6</sub></font>, <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R2">are_equipotent</a>  &amp; not <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="classes1.html#E4:5_1_1_5"><i><font color="Green">E18</font></i></a>, <a class="txt" href="classes1.html#E5:5_1_1_5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="classes1.html#E6:5_1_1_5"><i><font color="Green">E20</font></i></a>, <a class="ref" href="card_1.html#T44">CARD_1:44</a></i>;<br/>
<a NAME="E9:5_1_1_5"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="classes1.html#E15:5_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="classes1.html#E17:5_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="classes1.html#E7:5_1_1_5"><i><font color="Green">E21</font></i></a>, <a class="ref" href="classes1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E10:5_1_1_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:5_1_1"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <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="5_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:5_1_1_6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:5_1_1_6"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:5_1_1_6"/>
<font color="Maroon">c<sub>6</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="classes1.html#E8:5_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="classes1.html#E6:5_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="classes1.html#E19:5_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E22:5_1_1"/><b>then </b>
(  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E23:5_1_1"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div>
