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









<a NAME="E1:27"/><i><font color="Green">E21</font></i>: 
 <a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">A</font>,<font color="Maroon">G</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K22">B_SUP</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D10">BVFUNC_2:def 10</a></i>;<br/>
<a NAME="E2:27"/><i><font color="Green">E22</font></i>: 
 <a href="bvfunc_2.html#K6">All</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">B</font>,<font color="Maroon">G</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K21">B_INF</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D9">BVFUNC_2:def 9</a></i>;<br/>
<a NAME="E3:27"/><i><font color="Green">E25</font></i>: 
 <a href="bvfunc_2.html#K6">All</a> <font color="Maroon">a</font>,<font color="Maroon">B</font>,<font color="Maroon">G</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D9">BVFUNC_2:def 9</a></i>;<br/>
<a NAME="E4:27"/><i><font color="Green">E26</font></i>: 
 <a href="bvfunc_2.html#K7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">a</font>,<font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span>,<font color="Maroon">A</font>,<font color="Maroon">G</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K22">B_SUP</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">a</font>,<font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D10">BVFUNC_2:def 10</a></i>;<br/>
<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="E5:27"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<a NAME="E6:27"/><i><font color="Green">E36</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">CompF</a> <font color="Maroon">B</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">CompF</a> <font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">B</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="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <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">CompF</a> <font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span> &amp; not <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><a NAME="E2:27_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</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="bvfunc13.html#T2" target="_self">Th2</a>, <a class="ref" href="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>;<br/><b>hence </b><a NAME="E3:27_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bvfunc13.html#E4:27"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:27"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:27"/><b>then </b>
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</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#D5">VALUAT_1:def 5</a></i>;<br/>
<a NAME="E10:27"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <font color="Maroon">a</font>,<font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
;<br/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2"/>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <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">CompF</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><a NAME="E2:27_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">B_SUP</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</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="bvfunc_1.html#D20">BVFUNC_1:def 20</a></i>;<br/><b>hence </b><a NAME="E3:27_2"/>
contradiction
 <b>by </b><i><a class="txt" href="bvfunc13.html#E2:27"><i><font color="Green">E22</font></i></a>, <a class="txt" href="bvfunc13.html#E5:27"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:27"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <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">CompF</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">a</font>,<font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E1:27_3"/>
<font color="Maroon">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">CompF</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span>
 ;<br/>

<a NAME="E2:27_3"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc13.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:27_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span> &amp;  <a href="bvfunc_1.html#K20">EqClass</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">B</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="E4:27_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">a</font>,<font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i>, <a class="ref" href="bvfunc13.html#T4" target="_self">Th4</a>, <a class="ref" href="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>;<br/>
<b>hence </b><a NAME="E5:27_3"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">a</font>,<font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:27"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">B_SUP</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">a</font>,<font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5">CompF</a> <font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</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/>
<a NAME="E14:27"/><b>then </b>
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">a</font>,<font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span>,<font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</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="txt" href="bvfunc13.html#E3:27"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:27"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7">Ex</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K6">All</a> <font color="Maroon">a</font>,<font color="Maroon">B</font>,<font color="Maroon">G</font></span>)</span>,<font color="Maroon">A</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</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="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>


</div>
