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

<a NAME="E1:18_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> = <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> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>set </b><font color="Maroon">a</font> =  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>;<br/>
<a NAME="E3:18_1_1"/>
 <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>
 ;<br/>

<a NAME="E2:18_1_1_1_1"/><b>then </b>
 ex <font color="Olive">M</font>, <font color="Olive">N</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">N</font> )
 
<b>by </b><i><a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E3:18_1_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T108">XBOOLE_1:108</a></i>;<br/>
<b>hence </b><a NAME="E4:18_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1_1_1"/>
 <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">a</font> =  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon">ia</font> = <font color="Maroon">a</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">e</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E6:18_1_1"/><i><font color="Green">E42</font></i>: 
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">PA</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">PB</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<i><font color="Green">E43</font></i>:  <a href="setfam_1.html#K5">union</a> <font color="Maroon">ia</font> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">PA</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">PB</font></span>)</span>
<b>by </b><i><a class="ref" href="setfam_1.html#T39">SETFAM_1:39</a></i>
<br/>.= 
<font color="Maroon">Y</font>
<b>by </b><i/>
;<br/>
<a NAME="E8:18_1_1"/>
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ia</font> holds <br/>( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ia</font> or <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> or <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E1:18_1_1_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ia</font>
 ;<br/>

<a NAME="E2:18_1_1_3"/><b>then </b><i><font color="Green">E48</font></i>: 
not <font color="Maroon">A</font> <a href="hidden.html#R2">in</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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:18_1_1_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:18_1_1_3"/>
for <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ia</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_3_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font>, <font color="Olive">o</font>, <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>  holds  <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">n</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Olive">o</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">o</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">p</font></span>)</span> = 
<font color="Maroon">m</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">o</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/>
<b>hence </b><a NAME="E2:18_1_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">o</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E2:18_1_1_3_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ia</font>
 ;<br/>

<a NAME="E3:18_1_1_3_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">M</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">N</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:18_1_1_3_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font>
 <b>and </b><br/><a NAME="E6:18_1_1_3_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font>
 <b>and </b><br/><a NAME="E7:18_1_1_3_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">S</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">T</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:18_1_1_3_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font>
 <b>and </b><br/><a NAME="E10:18_1_1_3_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font>
 <b>and </b><br/><a NAME="E11:18_1_1_3_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E12:18_1_1_3_1"/>
( <font color="Maroon">M</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">T</font> or  not <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> or  not <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> )
 
;<br/>
<a NAME="E13:18_1_1_3_1"/><b>then </b>
( <font color="Maroon">M</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">T</font> or <font color="Maroon">M</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">S</font> or <font color="Maroon">N</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">T</font> )
 
<b>by </b><i>, , , , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E14:18_1_1_3_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">M</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">T</font> or <font color="Maroon">M</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">N</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">S</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">T</font></span>)</span> = 
<font color="Maroon">M</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">N</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">N</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E16:18_1_1_3_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:18_1_1_3"/>
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ia</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) ) )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:18_1_1"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font></span>)</span> <a href="subset_1.html#K6">\</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> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>


</div>
