<?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/>









<a NAME="E1:39"/><i><font color="Green">E16</font></i>: 
(  <a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a>  <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a>  <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/>
<b>assume </b><a NAME="E2:39"/>
<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="E3:39"/><b>then </b><i><font color="Green">E25</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/>
<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="E4:39"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" target="_self">All</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</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/>

<a NAME="E5:39"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21">B_INF</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</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> 
 
;<br/>
<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_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" target="_self">CompF</a> <font color="Maroon">PA</font>,<font color="Maroon">G</font></span>)</span> &amp; not <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</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="E2:39_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21">B_INF</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</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#D19">BVFUNC_1:def 19</a></i>;<br/><b>hence </b><a NAME="E3:39_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:39"/><i><font color="Green">E28</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="E8:39"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</a> <span class="p2">(<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></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"><span class="p2">(<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="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K10">'xor'</a> <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>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D8">BVFUNC_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_2"/>
( ( ( 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">u</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>  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>  ) ) 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">u</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; 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>  ) ) 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">u</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>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1"><b>suppose </b></a><a NAME="E1:39_2_1"/><i><font color="Green">E30</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">u</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>  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>  ) )
 ;<br/><div class="add"><a NAME="E2:39_2_1"/><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> 
 <b>by </b><i/>;<br/><a NAME="E3:39_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</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> 
 <b>by </b><i>, <a class="txt" href="bvfunc_2.html#E1:39"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E4:39_2_1"/><i><font color="Green">E33</font></i>: 
( <a href="margrel1.html#K7">FALSE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K8">TRUE</a>  <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K7">FALSE</a>  <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><a NAME="E5:39_2_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</a> <font color="Maroon">a</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>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K10">'xor'</a> <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>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D8">BVFUNC_1:def 8</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, , , </i>
;<br/><b>hence </b><a NAME="E7:39_2_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="bvfunc_1.html#K7">'xor'</a> <span class="p2">(<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></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_2.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_2"><b>suppose </b></a><a NAME="E1:39_2_2"/><i><font color="Green">E38</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">u</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; 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/><div class="add"><a NAME="E2:39_2_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<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="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/><b>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="E4:39_2_2"/><i><font color="Green">E48</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>  )
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E1:39"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E5:39_2_2"/><i><font color="Green">E50</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> 
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E1:39"><i><font color="Green">E16</font></i></a>, </i>;<br/><a NAME="E6:39_2_2"/><i><font color="Green">E51</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> <font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , , <a class="ref" href="bvfunc_1.html#D18">BVFUNC_1:def 18</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</a> <span class="p2">(<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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
<a href="margrel1.html#K8">TRUE</a>  <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, , , <a class="ref" href="bvfunc_2.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="ref" href="binarith.html#T7">BINARITH:7</a></i>
;<br/><b>hence </b><a NAME="E8:39_2_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"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</a> <span class="p2">(<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></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3"><b>suppose </b></a><a NAME="E1:39_2_3"/>
 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">u</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/><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="E3:39_2_3"/><i><font color="Green">E52</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">u</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="E4:39_2_3"/><i><font color="Green">E53</font></i>: 
( <a href="margrel1.html#K7">FALSE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K8">TRUE</a>  <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K7">FALSE</a>  <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_2_3_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>  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>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:39_2_3_1_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> 
 ;<br/><div class="add"><a NAME="E2:39_2_3_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<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="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="E3:39_2_3_1_1_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#E1:39"><i><font color="Green">E16</font></i></a>, <a class="ref" href="bvfunc_1.html#D18">BVFUNC_1:def 18</a></i>;<br/><a NAME="E4:39_2_3_1_1_1"/><b>then </b><i><font color="Green">E55</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#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E1:39"><i><font color="Green">E16</font></i></a>, <a class="ref" href="margrel1.html#T43">MARGREL1:43</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</a> <span class="p2">(<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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
<a href="margrel1.html#K7">FALSE</a>  <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i>, , , </i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="ref" href="binarith.html#T7">BINARITH:7</a></i>
;<br/><b>hence </b><a NAME="E6:39_2_3_1_1_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="bvfunc_1.html#K7">'xor'</a> <span class="p2">(<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></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:39_2_3_1_1_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" 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>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:39_2_3_1_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x2</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">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><a NAME="E4:39_2_3_1_1_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">x2</font>
 <b>by </b><i>, <a class="txt" href="bvfunc_2.html#E1:39"><i><font color="Green">E16</font></i></a>, , <a class="ref" href="bvfunc_1.html#D18">BVFUNC_1:def 18</a></i>;<br/><a NAME="E5:39_2_3_1_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_2.html#E1:39"><i><font color="Green">E16</font></i></a>, <a class="ref" href="margrel1.html#T43">MARGREL1:43</a></i>;<br/><a NAME="E6:39_2_3_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <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/><a NAME="E7:39_2_3_1_1_2"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span> <a href="xboolean.html#K10">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D8">BVFUNC_1:def 8</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, , , <a class="ref" href="bvfunc_2.html#T1" target="_self">Th1</a></i>
;<br/><b>hence </b><a NAME="E9:39_2_3_1_1_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"><font color="Maroon">u</font> <a href="bvfunc_1.html#K7">'xor'</a> <span class="p2">(<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></span>)</span>,<font color="Maroon">z</font>) <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:39_2_3"/>
<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="bvfunc_1.html#K7">'xor'</a> <span class="p2">(<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></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></div><b>end;</b></div>

</div>
