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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>) holds <br/> (  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )</span><br/>



<a NAME="E1:31"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
( (  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  <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>  ) implies  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  <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#K7" title="MARGREL1:func.7">FALSE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 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  <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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  <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#K7" title="MARGREL1:func.7">FALSE</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><br/>

<a NAME="E2:31_1"/>
<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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a></span>;<br/>
<a NAME="E3:31_1"/><span class="kw">then </span>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D19" title="MARGREL1:def.19">MARGREL1:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_1"/>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:31"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
( (  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  <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#K7" title="MARGREL1:func.7">FALSE</a>  ) implies  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  <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>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_2"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
 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  <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#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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  <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> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><br/>

<a NAME="E2:31_2"/>
<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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_2"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>;<br/>
<a NAME="E3:31_2"/><span class="kw">then </span>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D19" title="MARGREL1:def.19">MARGREL1:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:31_2"/>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E23">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:31_3"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
(  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  <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>  or  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  <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#K7" title="MARGREL1:func.7">FALSE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:31_3_1_1"/>
( ( (  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  <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;  not  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  <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#K7" title="MARGREL1:func.7">FALSE</a>  ) or ( (  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  <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#K7" title="MARGREL1:func.7">FALSE</a>  ) &amp;  not  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  <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>  ) or ( (  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  <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  <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#K7" title="MARGREL1:func.7">FALSE</a>  ) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_3"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:31_3_1_1_1"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
( (  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  <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;  not  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  <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#K7" title="MARGREL1:func.7">FALSE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )</span><br/><div class="add"><a NAME="E2:31_3_1_1_1"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
 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  <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#K7" title="MARGREL1:func.7">FALSE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><br/>

<a NAME="E1:31_3_1_1_1_1"/>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
;<br/>
<a NAME="E2:31_3_1_1_1_1"/><span class="kw">then </span>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_3_1_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_3_1_1_1_1"/>
<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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D19" title="MARGREL1:def.19">MARGREL1:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:31_3_1_1_1"/>
 <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_3_1_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><a NAME="E4:31_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A9</font></span>: 
 <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">O_el</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T2" target="_self" title="BVFUNC_1:th.2">Th5</a></span>;<br/><a NAME="E5:31_3_1_1_1"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">O_el</a> <font color="Maroon" title="c1">Y</font> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E1:31_3_1_1_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E6:31_3_1_1_1"/>
(  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_3_1_1_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E2:31_3_1_1_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a>, <a class="ref" href="bvfunc_1.html#T2" target="_self" title="BVFUNC_1:th.2">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:31_3_1_1_2"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
( (  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  <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#K7" title="MARGREL1:func.7">FALSE</a>  ) &amp;  not  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  <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )</span><br/><div class="add"><a NAME="E2:31_3_1_1_2"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
 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  <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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><br/>

<a NAME="E1:31_3_1_1_2_1"/>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
;<br/>
<a NAME="E2:31_3_1_1_2_1"/><span class="kw">then </span>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_3_1_1_2"><span class="lab"><font color="Green" title="E25">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_3_1_1_2_1"/>
<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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D19" title="MARGREL1:def.19">MARGREL1:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:31_3_1_1_2"/>
 <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">O_el</a> <font color="Maroon" title="c1">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_3_1_1_2"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><a NAME="E4:31_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A12</font></span>: 
 <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T2" target="_self" title="BVFUNC_1:th.2">Th5</a></span>;<br/><a NAME="E5:31_3_1_1_2"/>
(  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">O_el</a> <font color="Maroon" title="c1">Y</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E1:31_3_1_1_2"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E6:31_3_1_1_2"/>
(  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_3_1_1_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E2:31_3_1_1_2"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a>, <a class="ref" href="bvfunc_1.html#T2" target="_self" title="BVFUNC_1:th.2">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_1_3"><span class="kw">case </span></a><a NAME="E1:31_3_1_1_3"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
( (  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  <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  <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#K7" title="MARGREL1:func.7">FALSE</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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/> (  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )</span><br/><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )</span><br/><a NAME="E2:31_3_1_1_3"/>
<font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_3_1_1_3"><span class="lab"><font color="Green" title="E25">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:31_3_1_1_3"/>
(  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_3_1_1_3"><span class="lab"><font color="Green" title="E25">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:31_3"/>
(  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31_4"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
 not  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  <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> 
 <span class="kw">and </span><br/><a NAME="E2:31_4"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
 not  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  <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#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )</span><br/><a NAME="E3:31_4"/>
 <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">O_el</a> <font color="Maroon" title="c1">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_4"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><a NAME="E4:31_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A16</font></span>: 
 <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T2" target="_self" title="BVFUNC_1:th.2">Th5</a></span>;<br/><a NAME="E5:31_4"/>
(  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span> &amp;  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">O_el</a> <font color="Maroon" title="c1">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E2:31_4"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E6:31_4"/>
(  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E1:31_4"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E4:31_4"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a>, <a class="ref" href="bvfunc_1.html#T2" target="_self" title="BVFUNC_1:th.2">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:31"/>
(  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">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> &amp;  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
