<?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">B</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">C</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">D</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/>









<b>assume </b><a NAME="E1:7"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="enumset1.html#K1">}</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E2:7"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">C</font></span>)</span> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">D</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="7_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="E1:7_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">C</font></span>)</span> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:7_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> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">D</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="E3:7_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">D</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">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:7_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">C</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</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">d</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E6:7_1"/>
<font color="Maroon">a</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">B</font>,<font color="Maroon">C</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="E7:7_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B</font>,<font color="Maroon">C</font> &amp; 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/>
<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:7_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c</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">h</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span>;<br/>
<a NAME="E10:7_1"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">C</font>,<font color="Maroon">D</font></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:7_1"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="yellow14.html#T3">YELLOW14:3</a></i>;<br/>
<a NAME="E12:7_1"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:7_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:7_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:7_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1_1_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> )
 </span><b>by </b><i>, <a class="ref" href="bvfunc14.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1"><b>case </b></a><a NAME="E1:7_1_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="yellow14.html#T3">YELLOW14:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_2"><b>case </b></a><a NAME="E1:7_1_1_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_3"><b>case </b></a><a NAME="E1:7_1_1_1_1_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:7_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<i><font color="Green">E35</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> = 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T100">ENUMSET1:100</a></i>
;<br/>
<a NAME="E17:7_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></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="7_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:7_1_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_3_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> or <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> or <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> )
 </span><b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3_1_1_1"><b>case </b></a><a NAME="E1:7_1_3_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_3_1_1_1"/>
<font color="Maroon">t</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>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3_1_1_2"><b>case </b></a><a NAME="E1:7_1_3_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:7_1_3_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:7_1_3_1_1_2"/>
<font color="Maroon">t</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="7_1_3_1_1_3"><b>case </b></a><a NAME="E1:7_1_3_1_1_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:7_1_3_1_1_3"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:7_1_3_1_1_3"/>
<font color="Maroon">t</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="E3:7_1_3"/>
<font color="Maroon">t</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E19:7_1"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <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="bvfunc14.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E20:7_1"/><i><font color="Green">E38</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="7_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="E1:7_1_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:7_1_4"/>
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="7_1_4_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="E1:7_1_4_1"/><i><font color="Green">E47</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="7_1_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_4_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> )
 </span><b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_4_1_1_1_1"><b>case </b></a><a NAME="E1:7_1_4_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_4_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="txt" href="bvfunc14.html#E1:7"><i><font color="Green">E20</font></i></a>, <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="7_1_4_1_1_1_2"><b>case </b></a><a NAME="E1:7_1_4_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:7_1_4_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">d</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/><b>hence </b><a NAME="E3:7_1_4_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="txt" href="bvfunc14.html#E2:7"><i><font color="Green">E22</font></i></a>, <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="7_1_4_1_1_1_3"><b>case </b></a><a NAME="E1:7_1_4_1_1_1_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:7_1_4_1_1_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">d</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/><b>hence </b><a NAME="E3:7_1_4_1_1_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="txt" href="bvfunc14.html#E2:7"><i><font color="Green">E22</font></i></a>, <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="E3:7_1_4_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="E3:7_1_4"/><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="E4:7_1_4"/>
<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="E21:7_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="7_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:7_1_5"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:7_1_5"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">t</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E3:7_1_5"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p5">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span></span><a href="enumset1.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E4:7_1_5"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green">E20</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E5:7_1_5"/><b>then </b>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, , , <a class="ref" href="yellow14.html#T3">YELLOW14:3</a></i>;<br/>
<a NAME="E6:7_1_5"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c</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="E7:7_1_5"/>
<font color="Maroon">t</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="E22:7_1"/><b>then </b><i><font color="Green">E53</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="E23:7_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="E24:7_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="txt" href="bvfunc14.html#E1:7"><i><font color="Green">E20</font></i></a>, <a class="ref" href="bvfunc_2.html#D1">BVFUNC_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:7"/>
 <a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">C</font></span>)</span> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_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="E1:7_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="E3:7_2"/><i><font color="Green">E55</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="E4:7_2"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E5:7_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( <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> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">h</font> <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/>;<br/>
<a NAME="E6:7_2"/>
( <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> &amp; <font color="Maroon">C</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">D</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="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E7:7_2"/><b>then </b><i><font color="Green">E65</font></i>: 
( <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> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</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">D</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="E8:7_2"/><i><font color="Green">E66</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="E9:7_2"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><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> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span></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">D</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="7_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="E1:7_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><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> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span></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">D</font></span>)</span>
 ;<br/>

<a NAME="E2:7_2_1"/><b>then </b><i><font color="Green">E69</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">B</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">C</font></span>)</span> &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">D</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:7_2_1"/><i><font color="Green">E70</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="enumset1.html#K1">{</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">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_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="E1:7_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="E3:7_2_1_1"/><i><font color="Green">E71</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="7_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_2_1_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> or <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 </span><b>by </b><i>, , <a class="ref" href="bvfunc14.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:7_2_1_1_1_1_1"/>
<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="E2:7_2_1_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</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">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="bvfunc14.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:7_2_1_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_2_1_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</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">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="bvfunc14.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_1_1_1_3"><b>case </b></a><a NAME="E1:7_2_1_1_1_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_2_1_1_1_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</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">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="bvfunc14.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:7_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</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">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:7_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="7_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="E1:7_2_1_2"/><i><font color="Green">E72</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="7_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_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">B</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">C</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">D</font> )
 </span><b>by </b><i>, <a class="ref" href="bvfunc14.html#T1" target="_self">Th1</a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:7_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">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_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="7_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:7_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">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_2_1_1_3"><b>case </b></a><a NAME="E1:7_2_1_2_1_1_3"/>
<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">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_2_1_2_1_1_3"/>
<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="E3:7_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="E5:7_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="E6:7_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="E10:7_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><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> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span></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">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_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="E1:7_2_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:7_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="E3:7_2_2"/><b>then </b><i><font color="Green">E74</font></i>: 
( <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> &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">C</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">D</font> )
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E4:7_2_2"/><b>then </b>
<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">B</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">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:7_2_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><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> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span></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">D</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="E11:7_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><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> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span></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">D</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/>
<b>set </b><font color="Maroon">m</font> = <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="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">C</font></span>)</span>;<br/>
<a NAME="E12:7_2"/>
<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="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">C</font></span>)</span> <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="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E13:7_2"/><b>then </b><i><font color="Green">E76</font></i>: 
not <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="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">C</font></span>)</span> <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="E14:7_2"/>
<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="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">C</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E15:7_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">B</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">C</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B</font>,<font color="Maroon">C</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/>
<a NAME="E16:7_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">B</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">C</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="partit1.html#D4">PARTIT1:def 4</a></i>;<br/>
<a NAME="E17:7_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">D</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E18:7_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> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">D</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="E19:7_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">C</font></span>)</span> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">D</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="E4:7"/>
 <a href="bvfunc_2.html#K2">'/\'</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">C</font></span>)</span> <a href="partit1.html#K2">'/\'</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
