<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <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/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <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/> for <font color="Olive" title="b4">PA</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Olive" title="b3">b</font></span>)</span>,<font color="Olive" title="b4">PA</font>,<font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <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"><font color="Olive" title="b2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Olive" title="b3">b</font></span>)</span></span>)</span>,<font color="Olive" title="b4">PA</font>,<font color="Olive" title="b1">G</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <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/> for <font color="Olive" title="b3">PA</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Olive" title="b2">b</font></span>)</span>,<font color="Olive" title="b3">PA</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <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"><font color="Olive" title="b1">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Olive" title="b2">b</font></span>)</span></span>)</span>,<font color="Olive" title="b3">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">PA</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Olive" title="b1">PA</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <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"><font color="Maroon" title="c3">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>,<font color="Olive" title="b1">PA</font>,<font color="Maroon" title="c2">G</font></span>)</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <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"><font color="Maroon" title="c3">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span></span><br/>









<a NAME="E1:29"/><i><font color="Green" title="E4">A1</font></i>: 
 <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <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"><span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_2.html#K7" title="BVFUNC_2:func.7">Ex</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="bvfunc_2.html#T20" title="BVFUNC_2:th.20">BVFUNC_2:20</a></i>;<br/>
<a NAME="E2:29"/>
 <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#T16" title="BVFUNC_1:th.16">BVFUNC_1:16</a></i>;<br/>
<b>hence </b><a NAME="E3:29"/>
 <a href="bvfunc_2.html#K6" title="BVFUNC_2:func.6">All</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <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"><font color="Maroon" title="c3">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c2">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="bvfunc_3.html#T26" target="_self" title="BVFUNC_3:th.26">Th26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
