<?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">u</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">PA</font>, <font color="Maroon" title="c5">PB</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/>







<b>assume </b><a NAME="E1:22"/>
<font color="Maroon" title="c2">u</font> <a href="bvfunc_2.html#R3" title="BVFUNC_2:pred.3">is_independent_of</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font>
 ;<br/>

<a NAME="E2:22"/><b>then </b><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="bvfunc_1.html#R2" title="BVFUNC_1:pred.2">is_dependent_of</a>  <a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#D8" title="BVFUNC_2:def.8">BVFUNC_2:def 8</a></i>;<br/>
<a NAME="E3:22"/>
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <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> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</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">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<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>;<br/>

<a NAME="E1:22_1"/><i><font color="Green" title="E6">A2</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">PA</font>,<font color="Maroon" title="c3">G</font></span>)</span> &amp;  <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">PA</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font> )
 
<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="E2:22_1"/><i><font color="Green" title="E7">A3</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="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span> &amp;  <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">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font> )
 
<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="E3:22_1"/><i><font color="Green" title="E8">A4</font></i>: 
<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> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</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">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xboolean.html#K3" title="XBOOLEAN:func.3">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11" title="BVFUNC_1:def.11">BVFUNC_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1"/>
( <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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>  or <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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>  )
 </span><b>by </b><i><a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1"><b>case </b></a><a NAME="E1:22_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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/><div class="add"><b>hence </b><a NAME="E2:22_1_1_1_1"/>
<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> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</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">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="txt" href="bvfunc_4.html#E3:22_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_2"><b>case </b></a><a NAME="E1:22_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> 
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</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="bvfunc_4.html#E2:22_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="bvfunc_1.html#D20" title="BVFUNC_1:def.20">BVFUNC_1:def 20</a></i>;<br/><a NAME="E3:22_1_1_1_2"/><b>then </b>
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>  holds <br/>(  not <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="c4">PA</font>,<font color="Maroon" title="c3">G</font></span>)</span> or  not <font color="Maroon" title="c2">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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>  )
 <b>by </b><i><a class="txt" href="bvfunc_4.html#E2:22"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="bvfunc_4.html#E1:22_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="bvfunc_1.html#D18" title="BVFUNC_1:def.18">BVFUNC_1:def 18</a></i>;<br/><a NAME="E4:22_1_1_1_2"/><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">u</font>,<span class="p2">(<span class="default"><a href="bvfunc_2.html#K5" title="BVFUNC_2:func.5">CompF</a> <font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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/><b>hence </b><a NAME="E5:22_1_1_1_2"/>
<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> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</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">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="txt" href="bvfunc_4.html#E3:22_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:22_1"/>
<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> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</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">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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/>


</div><b>end;</b></div>
<a NAME="E4:22"/><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">u</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">I_el</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D14" title="BVFUNC_1:def.14">BVFUNC_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E5:22"/>
 <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">PA</font>,<font color="Maroon" title="c3">G</font> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c5">PB</font>,<font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#T19" title="BVFUNC_1:th.19">BVFUNC_1:19</a></i>;<br/>


</div>
