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

<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>thus </b><a NAME="E1:17"/>
 <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="cantor_1.html#T4" target="_self">Th4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>3</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="E2:17"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</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="E5:17"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="cantor_1.html#E2:17"><i><font color="Green">E14</font></i></a>, <a class="ref" href="cantor_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> );<br/>
<a NAME="E6:17"/><i><font color="Green">E16</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>5</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<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">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><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="E1:17_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>5</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</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="E4:17_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="cantor_1.html#E5:17"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cantor_1.html#E1:17_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="cantor_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E5:17_1"/>
( <font color="Maroon">c<sub>8</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="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>] )
 <b>by </b><i><a class="txt" href="cantor_1.html#E4:17_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>5</sub></font>, <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:17"/><i><font color="Green">E17</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>5</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="cantor_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="cantor_1.html#E6:17"><i><font color="Green">E16</font></i></a>);<br/></i>
<a NAME="E9:17"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="cantor_1.html#E5:17"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:17"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E11:17"/>
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="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </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="E1:17_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cantor_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:17_2"/>
<font color="Maroon">c<sub>8</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="cantor_1.html#E3:17_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="cantor_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E7:17_2"/>
<font color="Maroon">c<sub>9</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="cantor_1.html#E3:17_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:17_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>]
 
<b>by </b><i><a class="txt" href="cantor_1.html#E8:17"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:17_2"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="cantor_1.html#E3:17_2"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:17"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="cantor_1.html#E10:17"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>
<a NAME="E13:17"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E14:17"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E15:17"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T99">ZFMISC_1:99</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</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="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <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>9</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> ;<br/>
<a NAME="E18:17"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<a NAME="E1:17_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:17_3_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</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="E3:17_3_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cantor_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cantor_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="cantor_1.html#E8:17"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E4:17_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="cantor_1.html#E1:17_3_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:17_3_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="cantor_1.html#E3:17_3_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:17_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="cantor_1.html#E3:17_3_1"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:17_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:17_3_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</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="E3:17_3_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E4:17_3_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:17_3_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cantor_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="cantor_1.html#E4:17_3_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:17_3_2"/>
<font color="Maroon">c<sub>12</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="cantor_1.html#E6:17_3_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:17_3_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="cantor_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>]
 
<b>by </b><i><a class="txt" href="cantor_1.html#E8:17"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:17_3_2"/>
<font color="Maroon">c<sub>10</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="cantor_1.html#E3:17_3_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="cantor_1.html#E6:17_3_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="cantor_1.html#E1:17_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:17"/>
<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>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_4_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_4_1_1"><b>suppose </b></a><a NAME="E1:17_4_1_1"/><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:17_4_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E3:17_4_1_1"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T79">ZFMISC_1:79</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="cantor_1.html#E1:17_4_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cantor_1.html#E2:17_4_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/><a NAME="E6:17_4_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>11</sub></font> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4_1_1_1"><b>proof </b></a><div class="add">

<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>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:17_4_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 ;<br/><a NAME="E2:17_4_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:17_4_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>11</sub></font> : verum </span>}</span> 
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>12</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="E2:17_4_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>11</sub></font> : verum </span>}</span> 
 ;<br/>

<a NAME="E3:17_4_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>11</sub></font> st <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E4:17_4_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:17_4_1_1"/>
<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>8</sub></font>
 <b>by </b><i><a class="txt" href="cantor_1.html#E5:17"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cantor_1.html#E18:17"><i><font color="Green">E21</font></i></a>, <a class="ref" href="cantor_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_4_1_2"><b>suppose </b></a><a NAME="E1:17_4_1_2"/><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:17_4_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:17_4_1_2"/>
<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>8</sub></font>
 <b>by </b><i><a class="txt" href="cantor_1.html#E5:17"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cantor_1.html#E1:17_4_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E20:17"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cantor_1.html#E12:17"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cantor_1.html#E15:17"><i><font color="Green">E20</font></i></a>, <a class="ref" href="cantor_1.html#D4" target="_self">Def4</a></i>;<br/>


</div>
