<?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="E1:31"/><i><font color="Green">E16</font></i>: 
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="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">a</font>,<span class="p3">(<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></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_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/><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>  ) &amp;  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" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</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>  ) ) or ( ( 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/><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>  ) &amp; ( 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"><a href="valuat_1.html#K5">'not'</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>  ) ) ) or (  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" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; 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>  ) &amp;  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" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</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>  ) ) or (  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" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; 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>  ) &amp; ( 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"><a href="valuat_1.html#K5">'not'</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>  ) ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1"/><i><font color="Green">E25</font></i>: 
( ( 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/><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>  ) &amp;  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" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</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/><div class="add"><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="E3:31_1_1_1"/><i><font color="Green">E26</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"><a href="valuat_1.html#K5">'not'</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="E4:31_1_1_1"/>
 <a href="xboolean.html#K3">'not'</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="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/><a NAME="E5:31_1_1_1"/><b>then </b>
<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#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><a NAME="E6:31_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><b>hence </b><a NAME="E7:31_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">a</font>,<span class="p3">(<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></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_2"/><i><font color="Green">E27</font></i>: 
( ( 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/><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>  ) &amp; ( 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"><a href="valuat_1.html#K5">'not'</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/><div class="add"><a NAME="E2:31_1_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">a</font>,<span class="p2">(<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></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="bvfunc_1.html#D19">BVFUNC_1:def 19</a></i>;<br/><a NAME="E3:31_1_1_2"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<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></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/><a NAME="E4:31_1_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">a</font>,<span class="p3">(<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></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#K9">'not'</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/><b>hence </b><a NAME="E5:31_1_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">a</font>,<span class="p3">(<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></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_3"><b>suppose </b></a><a NAME="E1:31_1_1_3"/><i><font color="Green">E30</font></i>: 
(  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" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; 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>  ) &amp;  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" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</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/><div class="add"><a NAME="E2:31_1_1_3"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">a</font>,<span class="p2">(<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></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#D19">BVFUNC_1:def 19</a></i>;<br/><a NAME="E3:31_1_1_3"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<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></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="bvfunc_1.html#D20">BVFUNC_1:def 20</a></i>;<br/><a NAME="E4:31_1_1_3"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">a</font>,<span class="p3">(<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></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#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, <a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:31_1_1_3"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">a</font>,<span class="p3">(<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></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_4"><b>suppose </b></a><a NAME="E1:31_1_1_4"/><i><font color="Green">E33</font></i>: 
(  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" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; 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>  ) &amp; ( 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"><a href="valuat_1.html#K5">'not'</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/><div class="add"><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="E3:31_1_1_4"/><i><font color="Green">E36</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#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><a NAME="E4:31_1_1_4"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E5:31_1_1_4"/><b>then </b>
 <a href="xboolean.html#K3">'not'</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#NR2" target="_self">&lt;&gt;</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="E6:31_1_1_4"/><b>then </b>
 <a href="xboolean.html#K3">'not'</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#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T43">MARGREL1:43</a></i>;<br/><b>hence </b><a NAME="E7:31_1_1_4"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K21">B_INF</a> <font color="Maroon">a</font>,<span class="p3">(<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></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K22">B_SUP</a> <span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">k3</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:31"/><i><font color="Green">E38</font></i>: 
(  <a href="valuat_1.html#K5">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" target="_self">All</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k3</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k3</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">k4</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:31"/><i><font color="Green">E39</font></i>: 
(  <a href="bvfunc_2.html#K7" target="_self">Ex</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</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">k4</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k4</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:31"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">k3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E7:31"/>
 <a href="valuat_1.html#K5">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" target="_self">All</a> <font color="Maroon">a</font>,<font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bvfunc_2.html#K7" target="_self">Ex</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">PA</font>,<font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
