<?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="Red">::  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> <br/> for <font color="Olive" title="b2">d</font> being  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Olive" title="b2">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Olive" title="b2">d</font> )</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Olive" title="b1">d</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">d</font> be   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/>





<a NAME="E1:43"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T25" target="_self" title="BVFUNC_1:th.25">Th25</a></span>;<br/>
<a NAME="E2:43"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c3">d</font></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#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T25" target="_self" title="BVFUNC_1:th.25">Th25</a></span>;<br/>
<a NAME="E3:43"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c3">d</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T25" target="_self" title="BVFUNC_1:th.25">Th25</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E26">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E27">A5</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="c3">d</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="c3">d</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_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="E1:43_1_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="c3">d</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="c3">d</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="c3">d</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="c3">d</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="c3">d</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="c3">d</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="#E1:43_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:43_1_1_1_1"/><span class="lab"><font color="Green" title="E28">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="c3">d</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="c3">d</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E29">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_1_1_1_1_1"/><span class="lab"><font color="Green" title="E30">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  <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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1_1_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="E1:43_1_1_1_1_1_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#K7" title="MARGREL1:func.7">FALSE</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#K8" title="MARGREL1:func.8">TRUE</a>  ) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:43_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A9</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:43_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E32">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  <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="43_1_1_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="E1:43_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E1:43_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a></span>;<br/>
<a NAME="E2:43_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D21" title="MARGREL1:def.21">MARGREL1:def 21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_1_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="Red">::  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="E33">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A12</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="Red">::  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 contradiction</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:43_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="#E2:43_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_1_1_1_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:43_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E34">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  <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="43_1_1_1_1_1_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="E1:43_1_1_1_1_1_1_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c4">x</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="bvfunc_1.html#D7" target="_self" title="BVFUNC_1:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:43_1_1_1_1_1_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:43_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
(  <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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</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="#E1:43_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>;<br/><a NAME="E6:43_1_1_1_1_1_1_1_1"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c3">d</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c3">d</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="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E7:43_1_1_1_1_1_1_1_1"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:43_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E3:43_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E5:43_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="43_1_1_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:43_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A15</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:43_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A16</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="43_1_1_1_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="E1:43_1_1_1_1_1_1_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</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="c4">x</font></span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="margrel1.html#D21" title="MARGREL1:def.21">MARGREL1:def 21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:43_1_1_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:43_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A17</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="43_1_1_1_1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="E1:43_1_1_1_1_1_1_1_2_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c4">x</font></span>)</span> &amp; <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="bvfunc_1.html#D7" target="_self" title="BVFUNC_1:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:43_1_1_1_1_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:43_1_1_1_1_1_1_1_2"/>
 <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</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#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="#E1:43_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>;<br/><a NAME="E5:43_1_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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#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#T8" target="_self" title="BVFUNC_1:th.8">Th8</a></span>;<br/><a NAME="E6:43_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
 <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#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="#E1:43_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E7:43_1_1_1_1_1_1_1_2"/>
 <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c3">d</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="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E8:43_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="txt" href="#E1:43"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E6:43_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="bvfunc_1.html#T12" target="_self" title="BVFUNC_1:th.12">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_1_1_1_1_1_1_1_2"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E2:43_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E5:43_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="43_1_1_1_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:43_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E31">A20</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; ( 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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:43_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E32">A21</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="43_1_1_1_1_1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="E1:43_1_1_1_1_1_1_1_3_1"/>
( <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E1:43_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E31">A20</font></span></a></span>;<br/>
<a NAME="E2:43_1_1_1_1_1_1_1_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D21" title="MARGREL1:def.21">MARGREL1:def 21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_1_1_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:43_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E33">A22</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="43_1_1_1_1_1_1_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="E1:43_1_1_1_1_1_1_1_3_2"/>
( <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E1:43_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E31">A20</font></span></a></span>;<br/>
<a NAME="E2:43_1_1_1_1_1_1_1_3_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D7" target="_self" title="BVFUNC_1:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_1_1_1_1_1_1_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:43_1_1_1_1_1_1_1_3"/>
 <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</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#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="#E1:43_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E31">A20</font></span></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>;<br/><a NAME="E5:43_1_1_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A23</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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#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#T8" target="_self" title="BVFUNC_1:th.8">Th8</a></span>;<br/><a NAME="E6:43_1_1_1_1_1_1_1_3"/>
 <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c3">d</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="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E7:43_1_1_1_1_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="txt" href="#E1:43"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="bvfunc_1.html#T13" target="_self" title="BVFUNC_1:th.13">Th13</a></span>;<br/><span class="kw">hence </span><a NAME="E8:43_1_1_1_1_1_1_1_3"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E33">A22</font></span></a>, <a class="txt" href="#E2:43_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E32">A21</font></span></a>, <a class="txt" href="#E5:43_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E3:43_1_1_1_1_1"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="43_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_1_1_1_1_2"/><span class="lab"><font color="Green" title="E30">A24</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="E2:43_1_1_1_1_2"/>
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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><a NAME="E3:43_1_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A25</font></span>: 
 <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#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="#E1:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E30">A24</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E4:43_1_1_1_1_2"/>
 <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c3">d</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="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E5:43_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A26</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="txt" href="#E1:43"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A25</font></span></a>, <a class="ref" href="bvfunc_1.html#T12" target="_self" title="BVFUNC_1:th.12">Th12</a></span>;<br/><a NAME="E6:43_1_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A27</font></span>: 
<font color="Maroon" title="c3">d</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="#E3:43"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E7:43_1_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A28</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="43_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="E1:43_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="ref" href="bvfunc_1.html#T13" target="_self" title="BVFUNC_1:th.13">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:43_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c4">k4</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E9:43_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A29</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k4</font>
 <span class="kw">and </span><br/><a NAME="E10:43_1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A30</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">k4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">and </span><br/><a NAME="E11:43_1_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">k4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">k3</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E13:43_1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A31</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k3</font>
 <span class="kw">and </span><br/><a NAME="E14:43_1_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A32</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">and </span><br/><a NAME="E15:43_1_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">k3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/><a NAME="E16:43_1_1_1_1_2"/>
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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E1:43_1_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</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="c6">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="ref" href="margrel1.html#D21" title="MARGREL1:def.21">MARGREL1:def 21</a></span>;<br/>
<a NAME="E2:43_1_1_1_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</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="c6">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_1_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E17:43_1_1_1_1_2"/><span class="kw">then </span>
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="c5">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="c4">k4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A31</font></span></a>, <a class="txt" href="#E9:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A29</font></span></a></span>;<br/><a NAME="E18:43_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A33</font></span>: 
 <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A31</font></span></a>, <a class="txt" href="#E14:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A32</font></span></a>, <a class="txt" href="#E9:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="txt" href="#E10:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/><a NAME="E19:43_1_1_1_1_2"/>
 <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c3">d</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="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E20:43_1_1_1_1_2"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E7:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E18:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A33</font></span></a>, <a class="txt" href="#E5:43_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a>, <a class="ref" href="bvfunc_1.html#T9" target="_self" title="BVFUNC_1:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:43_1_1_1_1"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="43_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:43_1_1_1_2"/><span class="lab"><font color="Green" title="E28">A34</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="c3">d</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="c3">d</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E29">A35</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_1_1_1_2_1"/><span class="lab"><font color="Green" title="E30">A36</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_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="E1:43_1_1_1_2_1_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#K7" title="MARGREL1:func.7">FALSE</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#K8" title="MARGREL1:func.8">TRUE</a>  ) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1_1_2_1"><span class="lab"><font color="Green" title="E30">A36</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:43_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A37</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:43_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A38</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="43_1_1_1_2_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="E1:43_1_1_1_2_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c4">x</font></span>)</span> &amp; <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="bvfunc_1.html#D7" target="_self" title="BVFUNC_1:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:43_1_1_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="#E1:43_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E31">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:43_1_1_1_2_1_1_1_1"/>
 <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c3">d</font> <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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>;<br/><a NAME="E4:43_1_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A39</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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#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="#E2:43"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="bvfunc_1.html#T8" target="_self" title="BVFUNC_1:th.8">Th8</a></span>;<br/><a NAME="E5:43_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A40</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="43_1_1_1_2_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="E1:43_1_1_1_2_1_1_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</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="c4">x</font></span>)</span> &amp; <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="margrel1.html#D21" title="MARGREL1:def.21">MARGREL1:def 21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:43_1_1_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:43_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A41</font></span>: 
 <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="#E1:43_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E31">A37</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E7:43_1_1_1_2_1_1_1_1"/>
 <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c3">d</font> <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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E8:43_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="txt" href="#E1:43"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E6:43_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E35">A41</font></span></a>, <a class="ref" href="bvfunc_1.html#T12" target="_self" title="BVFUNC_1:th.12">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_1_1_1_2_1_1_1_1"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="txt" href="#E5:43_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E34">A40</font></span></a>, <a class="txt" href="#E4:43_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E33">A39</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="43_1_1_1_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:43_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A42</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:43_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A43</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="43_1_1_1_2_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="E1:43_1_1_1_2_1_1_1_2_1"/>
( <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="txt" href="#E1:43_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E31">A42</font></span></a></span>;<br/>
<a NAME="E2:43_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D7" target="_self" title="BVFUNC_1:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_1_1_1_2_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="Red">::  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="E33">A44</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_2_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_1_1_1_2_1_1_1_2_2"/><span class="lab"><font color="Green" title="E34">A45</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="Red">::  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 contradiction</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:43_1_1_1_2_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="#E2:43_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E32">A43</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_1_1_1_2_1_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1_1_2_1_1_1_2_2"><span class="lab"><font color="Green" title="E34">A45</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:43_1_1_1_2_1_1_1_2"/>
 <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c3">d</font> <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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>;<br/><a NAME="E5:43_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A46</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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#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="#E2:43"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="bvfunc_1.html#T8" target="_self" title="BVFUNC_1:th.8">Th8</a></span>;<br/><a NAME="E6:43_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A47</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="43_1_1_1_2_1_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="E1:43_1_1_1_2_1_1_1_2_3"/>
( <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="txt" href="#E1:43_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E31">A42</font></span></a></span>;<br/>
<a NAME="E2:43_1_1_1_2_1_1_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</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#D21" title="MARGREL1:def.21">MARGREL1:def 21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_1_1_1_2_1_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:43_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A48</font></span>: 
 <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#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="#E1:43_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E31">A42</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E8:43_1_1_1_2_1_1_1_2"/>
 <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c3">d</font> <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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_1_1_1_2_1_1_1_2"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:43_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E33">A44</font></span></a>, <a class="txt" href="#E6:43_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E35">A47</font></span></a>, <a class="txt" href="#E7:43_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E36">A48</font></span></a>, <a class="txt" href="#E5:43_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E34">A46</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="43_1_1_1_2_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:43_1_1_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E31">A49</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; ( 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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:43_1_1_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E32">A50</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="43_1_1_1_2_1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="E1:43_1_1_1_2_1_1_1_3_1"/>
( <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="txt" href="#E1:43_1_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E31">A49</font></span></a></span>;<br/>
<a NAME="E2:43_1_1_1_2_1_1_1_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D7" target="_self" title="BVFUNC_1:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_1_1_1_2_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c4">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:43_1_1_1_2_1_1_1_3"/>
 <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c3">d</font> <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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>;<br/><a NAME="E4:43_1_1_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A51</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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#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="#E2:43"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="bvfunc_1.html#T8" target="_self" title="BVFUNC_1:th.8">Th8</a></span>;<br/><a NAME="E5:43_1_1_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E34">A52</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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="43_1_1_1_2_1_1_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="E1:43_1_1_1_2_1_1_1_3_2"/>
( <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="txt" href="#E1:43_1_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E31">A49</font></span></a></span>;<br/>
<a NAME="E2:43_1_1_1_2_1_1_1_3_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</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#D21" title="MARGREL1:def.21">MARGREL1:def 21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_1_1_1_2_1_1_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:43_1_1_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E35">A53</font></span>: 
 <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="#E1:43_1_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E31">A49</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E7:43_1_1_1_2_1_1_1_3"/>
 <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c3">d</font> <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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E8:43_1_1_1_2_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="txt" href="#E1:43"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E6:43_1_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E35">A53</font></span></a>, <a class="ref" href="bvfunc_1.html#T12" target="_self" title="BVFUNC_1:th.12">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_1_1_1_2_1_1_1_3"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E32">A50</font></span></a>, <a class="txt" href="#E5:43_1_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E34">A52</font></span></a>, <a class="txt" href="#E4:43_1_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E33">A51</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E3:43_1_1_1_2_1"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="43_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:43_1_1_1_2_2"/>
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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="43_1_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="E1:43_1_1_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/>
<a NAME="E2:43_1_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T8" target="_self" title="BVFUNC_1:th.8">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_1_1_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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="c4">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="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E2:43_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A54</font></span>: 
 <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:43_1_1_1_2_2"/><span class="lab"><font color="Green" title="E31">A55</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="E4:43_1_1_1_2_2"/>
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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><a NAME="E5:43_1_1_1_2_2"/>
 <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <font color="Maroon" title="c3">d</font> <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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><a NAME="E6:43_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A56</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> <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> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="#E1:43"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:43_1_1_1_2_2"><span class="lab"><font color="Green" title="E31">A55</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">k4</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:43_1_1_1_2_2"/><span class="lab"><font color="Green" title="E33">A57</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k4</font>
 <span class="kw">and </span><br/><a NAME="E9:43_1_1_1_2_2"/><span class="lab"><font color="Green" title="E34">A58</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">k4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">and </span><br/><a NAME="E10:43_1_1_1_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">k4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">k3</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E12:43_1_1_1_2_2"/><span class="lab"><font color="Green" title="E35">A59</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k3</font>
 <span class="kw">and </span><br/><a NAME="E13:43_1_1_1_2_2"/><span class="lab"><font color="Green" title="E36">A60</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">and </span><br/><a NAME="E14:43_1_1_1_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">k3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/><a NAME="E15:43_1_1_1_2_2"/>
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"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E1:43_1_1_1_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c6">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D7" target="_self" title="BVFUNC_1:def.7">Def7</a></span>;<br/>
<a NAME="E2:43_1_1_1_2_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c6">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_1_1_1_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E16:43_1_1_1_2_2"/><span class="kw">then </span>
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="c5">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="c4">k4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_1_1_1_2_2"><span class="lab"><font color="Green" title="E35">A59</font></span></a>, <a class="txt" href="#E8:43_1_1_1_2_2"><span class="lab"><font color="Green" title="E33">A57</font></span></a></span>;<br/><a NAME="E17:43_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A61</font></span>: 
<font color="Maroon" title="c5">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1_1_1_2_2"><span class="lab"><font color="Green" title="E36">A60</font></span></a>, <a class="txt" href="#E9:43_1_1_1_2_2"><span class="lab"><font color="Green" title="E34">A58</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/><a NAME="E18:43_1_1_1_2_2"/>
 <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <font color="Maroon" title="c3">d</font> <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="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self" title="BVFUNC_1:def.17">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E19:43_1_1_1_2_2"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:43_1_1_1_2_2"><span class="lab"><font color="Green" title="E31">A55</font></span></a>, <a class="txt" href="#E12:43_1_1_1_2_2"><span class="lab"><font color="Green" title="E35">A59</font></span></a>, <a class="txt" href="#E8:43_1_1_1_2_2"><span class="lab"><font color="Green" title="E33">A57</font></span></a>, <a class="txt" href="#E17:43_1_1_1_2_2"><span class="lab"><font color="Green" title="E37">A61</font></span></a>, <a class="txt" href="#E2:43_1_1_1_2_2"><span class="lab"><font color="Green" title="E30">A54</font></span></a>, <a class="txt" href="#E6:43_1_1_1_2_2"><span class="lab"><font color="Green" title="E32">A56</font></span></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self" title="BVFUNC_1:def.16">Def16</a>, <a class="ref" href="bvfunc_1.html#T8" target="_self" title="BVFUNC_1:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:43_1_1_1_2"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1_1_2"><span class="lab"><font color="Green" title="E29">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="43_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:43_1_1_1_3"/><span class="lab"><font color="Green" title="E28">A62</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="c3">d</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="c3">d</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="Red">::  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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><a NAME="E2:43_1_1_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:43_1_1_1_3"><span class="lab"><font color="Green" title="E28">A62</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_1_1_1_3"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1_1_3"><span class="lab"><font color="Green" title="E28">A62</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E3:43_1"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="43_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_2"/><span class="lab"><font color="Green" title="E27">A63</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="c3">d</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="E2:43_2"/><span class="lab"><font color="Green" title="E28">A64</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="c3">d</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_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="E1:43_2_1_1"/>
( <font color="Maroon" title="c3">d</font> <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> or <font color="Maroon" title="c3">d</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><span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T24" target="_self" title="BVFUNC_1:th.24">Th24</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:43_2_1_1_1"/>
<font color="Maroon" title="c3">d</font> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:43_2_1_1_1"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E28">A64</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="43_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:43_2_1_1_2"/>
<font color="Maroon" title="c3">d</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:43_2_1_1_2"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E27">A63</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E4:43_2"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:43"/>
(  <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</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> &amp;  <a href="bvfunc_1.html#K18" title="BVFUNC_1:func.18">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="bvfunc_1.html#K2" title="BVFUNC_1:func.2">'&amp;'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E26">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
