<?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="partit1.html#K1">PARTITIONS</a> <font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">u</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">PA</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/>









<b>assume </b><a NAME="E1:42"/>
<font color="Maroon">u</font> <a href="bvfunc_2.html#R3" target="_self">is_independent_of</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:42"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">u</font> <a href="bvfunc_1.html#R2">is_dependent_of</a>  <a href="bvfunc_2.html#K5" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:42"/><i><font color="Green">E25</font></i>: 
 <a href="bvfunc_2.html#K7" target="_self">Ex</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font>,<font color="Maroon">G</font> <a href="bvfunc_1.html#R1">'&lt;'</a> <font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="bvfunc_1.html#D15">BVFUNC_1:def 15</a><br/>

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><a NAME="E2:42_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D20">BVFUNC_1:def 20</a></i>;<br/><b>hence </b><a NAME="E3:42_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:42_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/>
<a NAME="E5:42_1"/>
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span> <a href="xboolean.html#K4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_2.html#E2:42"><i><font color="Green">E16</font></i></a>, <a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>;<br/>
<a NAME="E6:42_1"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/>
<a NAME="E7:42_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp;  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K5" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>
<a NAME="E8:42_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_2.html#E2:42"><i><font color="Green">E16</font></i></a>, , <a class="ref" href="bvfunc_1.html#D20">BVFUNC_1:def 20</a></i>;<br/>
<a NAME="E9:42_1"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="txt" href="bvfunc_2.html#E2:42"><i><font color="Green">E16</font></i></a>, , <a class="ref" href="bvfunc_1.html#D18">BVFUNC_1:def 18</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
<a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i>, , <a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 

;<br/>
<b>hence </b><a NAME="E11:42_1"/>
<a href="funct_2.html#K3">K121</a>(<font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ,<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">z</font>) <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:42"/>
<font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="bvfunc_1.html#R1">'&lt;'</a>  <a href="bvfunc_2.html#K7" target="_self">Ex</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font>,<font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="bvfunc_1.html#D15">BVFUNC_1:def 15</a><br/>

<b>assume </b><a NAME="E1:42_2"/>
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<a NAME="E2:42_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K4">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>;<br/>
<a NAME="E3:42_2"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/>
<a NAME="E4:42_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp;  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K5" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="t_1topsp.html#D1">T_1TOPSP:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_2_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> or  not <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><a NAME="E2:42_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D20">BVFUNC_1:def 20</a></i>;<br/><b>hence </b><a NAME="E3:42_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E7:42_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/>
<a NAME="E8:42_2"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="txt" href="bvfunc_2.html#E2:42"><i><font color="Green">E16</font></i></a>, , <a class="ref" href="bvfunc_1.html#D18">BVFUNC_1:def 18</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> = 
<a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i>, , <a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 

;<br/>
<b>hence </b><a NAME="E10:42_2"/>
<a href="funct_2.html#K3">K121</a>(<font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span>,<font color="Maroon">z</font>) <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#D20">BVFUNC_1:def 20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:42"/>
 <a href="bvfunc_2.html#K7" target="_self">Ex</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#T18">BVFUNC_1:18</a></i>;<br/>


</div>
