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

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

<b>assume </b><a NAME="E1:2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">B</font> is <a href="finsub_1.html#V2">(B2)</a>
 ;<br/>

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E3:2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">sf</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>  st <font color="Olive">sf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">sf</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>;<br/>
<a NAME="E4:2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E5:2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E2:2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E3:2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">b</font>]
 ;<br/><b>thus </b><a NAME="E4:2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">b</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:2_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">sf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">sx</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E3:2_1_1"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fb</font> = <font color="Maroon">b</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E5:2_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fx</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E7:2_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">fb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
 <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><font color="Maroon">fb</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">fx</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">fb</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">fx</font></span>)</span>
<b>by </b><i><a class="ref" href="mssubfam.html#T8">MSSUBFAM:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">fb</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">sx</font>
<b>by </b><i><a class="ref" href="mssubfam.html#T9">MSSUBFAM:9</a></i>
;<br/>
<b>hence </b><a NAME="E9:2_1_1"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">sf</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, , , , , <a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">S</font>]
 
<b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(, , );<br/></i>
<b>hence </b><a NAME="E8:2"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>


</div>
