<?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">a</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="b2">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_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">PA</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b2">PA</font></span><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> ; <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_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">PA</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Olive" title="b1">PA</font></span><br/><b>let </b><font color="Maroon" title="c3">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_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">PA</font></span><br/>





<a NAME="E1:54"/><i><font color="Green" title="E26">A1</font></i>: 
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> holds  <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#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> <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#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span><br/>

<div><i><font color="Green" title="E27">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1_1"/><i><font color="Green" title="E28">A3</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#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">PA</font> 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#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp;  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#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">PA</font> &amp; <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</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>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c5">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="E3:54_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">PA</font> &amp; <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</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="E4:54_1_1"/>
<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="c2">a</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="#E3:54_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="margrel1.html#D20" title="MARGREL1:def.20">MARGREL1:def 20</a></i>;<br/><b>hence </b><a NAME="E5:54_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:54_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="#E3:54_1_1"><i><font color="Green" title="E29">A4</font></i></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><b>end;</b></div>
<div><i><font color="Green" title="E28">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1_2"/><i><font color="Green" title="E29">A6</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#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">PA</font> 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#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp; ( 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#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">PA</font> or  not <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</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>  ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span><br/><a NAME="E2:54_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</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" target="_self" title="BVFUNC_1:def.19">Def19</a></i>;<br/><a NAME="E3:54_1_2"/><b>then </b>
<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#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</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/><b>hence </b><a NAME="E4:54_1_2"/>
<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#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="#E1:54_1_2"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="bvfunc_1.html#D20" target="_self" title="BVFUNC_1:def.20">Def20</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><b>end;</b></div>
<div><i><font color="Green" title="E29">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1_3"/><i><font color="Green" title="E30">A8</font></i>: 
(  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#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">PA</font> &amp; not <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#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp;  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#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">PA</font> &amp; <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</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>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span><br/><a NAME="E2:54_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</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" target="_self" title="BVFUNC_1:def.19">Def19</a></i>;<br/><a NAME="E3:54_1_3"/><b>then </b>
<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#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#D20" title="MARGREL1:def.20">MARGREL1:def 20</a></i>;<br/><b>hence </b><a NAME="E4:54_1_3"/>
<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#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="#E1:54_1_3"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="bvfunc_1.html#D20" target="_self" title="BVFUNC_1:def.20">Def20</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><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1_4"/><i><font color="Green" title="E30">A9</font></i>: 
(  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#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">PA</font> &amp; not <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#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp; ( 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#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">PA</font> or  not <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</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>  ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c5">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="E3:54_1_4"/><i><font color="Green" title="E31">A10</font></i>: 
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">PA</font> &amp; <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">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="E4:54_1_4"/>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">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="#E3:54_1_4"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/><a NAME="E5:54_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#D20" title="MARGREL1:def.20">MARGREL1:def 20</a></i>;<br/><b>hence </b><a NAME="E6:54_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:54_1_4"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="#E3:54_1_4"><i><font color="Green" title="E31">A10</font></i></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><b>end;</b></div>
<b>hence </b><a NAME="E5:54_1"/>
<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#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="#E1:54_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="#E2:54_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="#E3:54_1"><i><font color="Green" title="E29">A7</font></i></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><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">k3</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:54"/><i><font color="Green" title="E27">A11</font></i>: 
(  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k3</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">k3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">k4</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:54"/><i><font color="Green" title="E28">A12</font></i>: 
(  <a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">PA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k4</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">k4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">k4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:54"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> holds <br/><font color="Maroon" title="c4">k3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font>
 
<b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E3:54"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="#E5:54"><i><font color="Green" title="E28">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:54"/>
 <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">B_INF</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">PA</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K21" title="BVFUNC_1:func.21">B_SUP</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">PA</font>
 <b>by </b><i><a class="txt" href="#E3:54"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="#E5:54"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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>
