<?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">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/>







<a NAME="E1:23"/>
for <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16">O_el</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_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>;<br/>

<a NAME="E1:23_1"/>
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" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="bvfunc_1.html#K16">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</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="23_1_1"><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="E1:23_1_1"/>
<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" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span>
 ;<br/>

<a NAME="E2:23_1_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K16">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</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="bvfunc_1.html#D13">BVFUNC_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:23_1_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K16">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</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>
<b>hence </b><a NAME="E2:23_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" target="_self">Ex</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16">O_el</a> <font color="Maroon">Y</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:23"/>
 <a href="bvfunc_2.html#K7" target="_self">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16">O_el</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">PA</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K16">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D13">BVFUNC_1:def 13</a></i>;<br/>


</div>
