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

<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1">PARTITIONS</a> <font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/>







<b>assume </b><a NAME="E1:2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E2:2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">A</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:2_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">A</font>,<font color="Maroon">B</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>
 
<b>by </b><i><a class="ref" href="partit1.html#D4">PARTIT1:def 4</a></i>;<br/>
<a NAME="E3:2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; not <font color="Maroon">x</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/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:2_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">h0</font> = <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<a NAME="E6:2_1"/><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<a NAME="E7:2_1"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_4.html#T67">FUNCT_4:67</a></i>;<br/>
<a NAME="E8:2_1"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_4.html#T67">FUNCT_4:67</a></i>;<br/>
<a NAME="E9:2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <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="2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:2_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_1"><b>case </b></a><a NAME="E1:2_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_2"><b>case </b></a><a NAME="E1:2_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_1_1"/>
<font color="Maroon">y</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>then reconsider </b><font color="Maroon">F</font> =  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E11:2_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:2_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_2_1_1"/>
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> or <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2_1_1_1"><b>case </b></a><a NAME="E1:2_1_2_1_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2_1_1_2"><b>case </b></a><a NAME="E1:2_1_2_1_1_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:2_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</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="E1:2_1_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:2_1_3"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:2_1_3_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_3_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_3_1_1_1_1"><b>case </b></a><a NAME="E1:2_1_3_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_3_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_3_1_1_1_2"><b>case </b></a><a NAME="E1:2_1_3_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_3_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_1_3_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:2_1_3"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:2_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:2_1"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</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="E1:2_1_4"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:2_1_4"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:2_1_4"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="funct_4.html#T67">FUNCT_4:67</a></i>;<br/>
<a NAME="E4:2_1_4"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E5:2_1_4"/><b>then </b>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:2_1_4"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</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="E14:2_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E15:2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E16:2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">G</font>
 <b>by </b><i>, , , , <a class="ref" href="bvfunc_2.html#D1">BVFUNC_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:2"/>
 <a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">F</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:2_2"/><i><font color="Green">E39</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">d</font> ) &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="bvfunc_2.html#D1">BVFUNC_2:def 1</a></i>;<br/>
<a NAME="E4:2_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:2_2"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:2_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E7:2_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:2_2"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">x</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:2_2"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</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="E1:2_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<a NAME="E2:2_2_1"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</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="E1:2_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> or <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 </span><b>by </b><i>, , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:2_2_1_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:2_2_1_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:2_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:2_2_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/><font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:2_2_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:2_2_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:2_2_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2_2_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:2_2_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:2_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:2_2"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</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="E1:2_2_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:2_2_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E3:2_2_2"/><b>then </b>
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:2_2_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span>
 <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="E11:2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E12:2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E13:2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">A</font>,<font color="Maroon">B</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>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="partit1.html#D4">PARTIT1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:2"/>
 <a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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