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<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>, <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><b>that </b><br/><a NAME="E1:5"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">G</font> is <a href="bvfunc_2.html#V2">independent</a>
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green">E26</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/>

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

<b>assume </b><a NAME="E3:5"/><i><font color="Green">E27</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> )
 ;<br/>

<b>set </b><font color="Maroon">h2</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="E4:5"/>
<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> <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="5_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="E1:5_1"/>
<font color="Maroon">x</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>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:5_1"/>
<font color="Maroon">x</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>
<b>then reconsider </b><font color="Maroon">SS</font> = <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> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E6:5"/><i><font color="Green">E31</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> &amp;  <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> <font color="Maroon">SS</font> )
 
<b>by </b><i>, <a class="ref" href="funct_4.html#T65">FUNCT_4:65</a>, <a class="ref" href="funct_4.html#T67">FUNCT_4:67</a></i>;<br/>
<a NAME="E7:5"/>
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="5_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="E1:5_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:5_2"/><b>then </b>
( <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> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:5_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>
<a NAME="E8:5"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">SS</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#D5">BVFUNC_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:5"/>
<font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">b</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="mssubfam.html#T10">MSSUBFAM:10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>


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