<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon" title="c3">G</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>;<br/>









<a NAME="E1:8"/><i><font color="Green" title="E8">A1</font></i>: 
 <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K22" title="BVFUNC_1:func.22">B_SUP</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D10" title="BVFUNC_2:def.10">BVFUNC_2:def 10</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="bvfunc_1.html#D15" title="BVFUNC_1:def.15">BVFUNC_1:def 15</a><br/>

<b>assume </b><a NAME="E2:8"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/>

<a NAME="E3:8"/><b>then </b><i><font color="Green" title="E9">A2</font></i>: 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="margrel1.html#D20" title="MARGREL1:def.20">MARGREL1:def 20</a></i>;<br/>
<div><i><font color="Green" title="E10">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 ;<br/><a NAME="E2:8_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K22" title="BVFUNC_1:func.22">B_SUP</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D20" title="BVFUNC_1:def.20">BVFUNC_1:def 20</a></i>;<br/><a NAME="E3:8_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_2.html#D10" title="BVFUNC_2:def.10">BVFUNC_2:def 10</a></i>;<br/><b>hence </b><a NAME="E4:8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bvfunc13.html#E3:8"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:8"/><i><font color="Green" title="E11">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span> holds <br/>for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span> holds <br/><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>;<br/>

<b>assume </b><a NAME="E1:8_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 ;<br/>

<a NAME="E2:8_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc13.html#E4:8"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:8_2"/>
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span> holds <br/><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc13.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="bvfunc_1.html#D20" title="BVFUNC_1:def.20">BVFUNC_1:def 20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:8"/><i><font color="Green" title="E12">A5</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<a NAME="E7:8"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span> holds <br/><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>;<br/>

<b>assume </b><a NAME="E1:8_3"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">z</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 ;<br/>

<a NAME="E2:8_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E3:8_3"/>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc13.html#E5:8"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="bvfunc13.html#E1:8_3"><i><font color="Green" title="E13">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:8"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K22" title="BVFUNC_1:func.22">B_SUP</a> <font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D20" title="BVFUNC_1:def.20">BVFUNC_1:def 20</a></i>;<br/>
<a NAME="E9:8"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D10" title="BVFUNC_2:def.10">BVFUNC_2:def 10</a></i>;<br/>
<a NAME="E10:8"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc13.html#E6:8"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="bvfunc_1.html#D19" title="BVFUNC_1:def.19">BVFUNC_1:def 19</a></i>;<br/>
<a NAME="E11:8"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D9" title="BVFUNC_2:def.9">BVFUNC_2:def 9</a></i>;<br/>
<a NAME="E12:8"/><b>then </b>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
;<br/>
<b>hence </b><a NAME="E13:8"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p3">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#D20" title="MARGREL1:def.20">MARGREL1:def 20</a></i>;<br/>


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