<?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>, <font color="Maroon" title="c3">b</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="c4">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="c5">PA</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>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="E1:24"/><i><font color="Green" title="E5">A1</font></i>: 
<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"><font color="Maroon" title="c2">a</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">'&amp;'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">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/>

<a NAME="E2:24"/><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="c5">PA</font>,<font color="Maroon" title="c4">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">PA</font>,<font color="Maroon" title="c4">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">PA</font>,<font color="Maroon" title="c4">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/>
<div><i><font color="Green" title="E7">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_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">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">'&amp;'</a> <font color="Maroon" title="c3">b</font></span>)</span> <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>  )
 ;<br/><a NAME="E2:24_1"/><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"><font color="Maroon" title="c2">a</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">'&amp;'</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">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> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D19" title="BVFUNC_1:def.19">BVFUNC_1:def 19</a></i>;<br/><b>hence </b><a NAME="E3:24_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bvfunc_4.html#E1:24"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_2"/>
 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">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> &amp; not <font color="Maroon" title="c3">b</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>  )
 ;<br/><b>then consider </b><font color="Maroon" title="c7">x1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:24_2"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c7">x1</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">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 ;<br/><a NAME="E4:24_2"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</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="bvfunc_4.html#E3:24_2"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/><a NAME="E5:24_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">'&amp;'</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</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#E3:24"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="bvfunc_4.html#E3:24_2"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/><a NAME="E6:24_2"/><b>then </b><i><font color="Green" title="E10">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</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="valuat_1.html#D6" title="VALUAT_1:def.6">VALUAT_1:def 6</a></i>;<br/><a NAME="E7:24_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <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="bvfunc_4.html#E4:24_2"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E8:24_2"/>
contradiction
 <b>by </b><i><a class="txt" href="bvfunc_4.html#E6:24_2"><i><font color="Green" title="E10">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:24"/><b>then </b><i><font color="Green" title="E8">A7</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_INF</a> <font color="Maroon" title="c3">b</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">PA</font>,<font color="Maroon" title="c4">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="ref" href="bvfunc_1.html#D19" title="BVFUNC_1:def.19">BVFUNC_1:def 19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_3"/>
 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">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> &amp; not <font color="Maroon" title="c2">a</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>  )
 ;<br/><b>then consider </b><font color="Maroon" title="c7">x1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:24_3"/><i><font color="Green" title="E9">A8</font></i>: 
( <font color="Maroon" title="c7">x1</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">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 ;<br/><a NAME="E4:24_3"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</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="bvfunc_4.html#E3:24_3"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/><a NAME="E5:24_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">'&amp;'</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</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#E3:24"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="bvfunc_4.html#E3:24_3"><i><font color="Green" title="E9">A8</font></i></a></i>;<br/><a NAME="E6:24_3"/><b>then </b><i><font color="Green" title="E11">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</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="valuat_1.html#D6" title="VALUAT_1:def.6">VALUAT_1:def 6</a></i>;<br/><a NAME="E7:24_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <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="bvfunc_4.html#E4:24_3"><i><font color="Green" title="E10">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E8:24_3"/>
contradiction
 <b>by </b><i><a class="txt" href="bvfunc_4.html#E6:24_3"><i><font color="Green" title="E11">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:24"/><b>then </b>
<font color="Maroon" title="c2">a</font> <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#E2:24"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">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="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<b>by </b><i><a class="txt" href="bvfunc_4.html#E5:24"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="valuat_1.html#D6" title="VALUAT_1:def.6">VALUAT_1:def 6</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 

;<br/>
<b>hence </b><a NAME="E9:24"/>
<a href="funct_2.html#K3" title="FUNCT_2:func.3">K69</a>(<font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">'&amp;'</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c4">G</font></span>)</span></span>)</span>,<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>
