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

<span class="kw">let </span><font color="Maroon" title="c1">Y</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>)<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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Olive" title="b2">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#K13" title="BVFUNC_1:func.13">'imp'</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="firebrick">::  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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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="firebrick">::  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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">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="E2:39"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k3</font>
 <span class="kw">and </span><br/><a NAME="E3:39"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">and </span><br/><a NAME="E4:39"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">k3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 <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">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="E6:39"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k4</font>
 <span class="kw">and </span><br/><a NAME="E7:39"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">k4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">and </span><br/><a NAME="E8:39"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">k4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 <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="E9:39"/>
 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"><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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <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></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> <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="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><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="firebrick">::  thesis: </font></span></a><span class="hide"> <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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <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></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"><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><br/>

<a NAME="E1:39_1"/>
<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="c6">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#D11" target="_self" title="BVFUNC_1:def.11">Def14</a></span>;<br/>
<a NAME="E2:39_1"/><span class="kw">then </span>
<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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <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></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"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a></span>)</span> <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#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_1"/>
<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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <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></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"><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 class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D11" target="_self" title="BVFUNC_1:def.11">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:39"/><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="c4">k3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/>
<a NAME="E11:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#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="#E2:39"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/>
<a NAME="E12:39"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> <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#K13" title="BVFUNC_1:func.13">'imp'</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T22" target="_self" title="BVFUNC_1:th.22">Th25</a></span>;<br/>
<a NAME="E13:39"/><span class="lab"><font color="Green" title="E29">A7</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#T22" target="_self" title="BVFUNC_1:th.22">Th25</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">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="E15:39"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
 <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k4</font>
 <span class="kw">and </span><br/><a NAME="E16:39"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">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="E17:39"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">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="c7">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="E19:39"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<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#K13" title="BVFUNC_1:func.13">'imp'</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k3</font>
 <span class="kw">and </span><br/><a NAME="E20:39"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">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="E21:39"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">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="E22:39"/>
 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"><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#K13" title="BVFUNC_1:func.13">'imp'</a> <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></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> <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="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><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#K13" title="BVFUNC_1:func.13">'imp'</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">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="c8">x</font></span><br/>

<a NAME="E1:39_2"/>
( <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#K13" title="BVFUNC_1:func.13">'imp'</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<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="c8">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c8">x</font></span>)</span> &amp; <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="c8">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#D8" target="_self" title="BVFUNC_1:def.8">Def11</a>, <a class="ref" href="bvfunc_1.html#D11" target="_self" title="BVFUNC_1:def.11">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:39_2"/>
<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#K13" title="BVFUNC_1:func.13">'imp'</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">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="c8">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:39"/><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="c7">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="c6">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="#E19:39"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E30">A8</font></span></a></span>;<br/>
<a NAME="E24:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A12</font></span>: 
<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#K13" title="BVFUNC_1:func.13">'imp'</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#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="#E19:39"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E20:39"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E16:39"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">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="E26:39"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
 <a href="bvfunc_1.html#K15" title="BVFUNC_1:func.15">O_el</a> <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k4</font>
 <span class="kw">and </span><br/><a NAME="E27:39"/><span class="lab"><font color="Green" title="E36">A14</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">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="E28:39"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">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="c9">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="E30:39"/><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k3</font>
 <span class="kw">and </span><br/><a NAME="E31:39"/><span class="lab"><font color="Green" title="E38">A16</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">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="E32:39"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">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="E33:39"/>
 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"><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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <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></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> <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="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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="c10">x</font></span><br/>

<a NAME="E1:39_3"/>
<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="c10">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#D10" target="_self" title="BVFUNC_1:def.10">Def13</a></span>;<br/>
<a NAME="E2:39_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A17</font></span>: 
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<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="c10">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a></span>)</span> <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#D11" target="_self" title="BVFUNC_1:def.11">Def14</a></span>;<br/>
<a NAME="E3:39_3"/>
<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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<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="c10">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:39_3"/>
<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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_3"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="bvfunc_1.html#D10" target="_self" title="BVFUNC_1:def.10">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E34:39"/><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="c9">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="c8">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="#E30:39"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E26:39"><span class="lab"><font color="Green" title="E35">A13</font></span></a></span>;<br/>
<a NAME="E35:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A18</font></span>: 
<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="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#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="#E30:39"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E31:39"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="txt" href="#E26:39"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E27:39"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">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="E37:39"/><span class="lab"><font color="Green" title="E40">A19</font></span>: 
 <a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k4</font>
 <span class="kw">and </span><br/><a NAME="E38:39"/><span class="lab"><font color="Green" title="E41">A20</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">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="E39:39"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">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="c11">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="E41:39"/><span class="lab"><font color="Green" title="E42">A21</font></span>: 
<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#K13" title="BVFUNC_1:func.13">'imp'</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k3</font>
 <span class="kw">and </span><br/><a NAME="E42:39"/><span class="lab"><font color="Green" title="E43">A22</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">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="E43:39"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">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="E44:39"/>
 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"><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#K13" title="BVFUNC_1:func.13">'imp'</a> <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></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> <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="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><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#K13" title="BVFUNC_1:func.13">'imp'</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c12">x</font></span><br/>

<a NAME="E1:39_4"/>
<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="c12">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#D10" target="_self" title="BVFUNC_1:def.10">Def13</a></span>;<br/>
<a NAME="E2:39_4"/><span class="kw">then </span>
<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#K13" title="BVFUNC_1:func.13">'imp'</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_4"/>
<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#K13" title="BVFUNC_1:func.13">'imp'</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D11" target="_self" title="BVFUNC_1:def.11">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E45:39"/><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="c11">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="c10">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="#E41:39"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="txt" href="#E37:39"><span class="lab"><font color="Green" title="E40">A19</font></span></a></span>;<br/>
<a NAME="E46:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A23</font></span>: 
<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#K13" title="BVFUNC_1:func.13">'imp'</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#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="#E41:39"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="txt" href="#E42:39"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E37:39"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E38:39"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/>
<a NAME="E47:39"/><span class="lab"><font color="Green" title="E45">A24</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#T22" target="_self" title="BVFUNC_1:th.22">Th25</a></span>;<br/>
<a NAME="E48:39"/><span class="lab"><font color="Green" title="E46">A25</font></span>: 
<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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> <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#K13" title="BVFUNC_1:func.13">'imp'</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T22" target="_self" title="BVFUNC_1:th.22">Th25</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E47">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_5"/><span class="lab"><font color="Green" title="E48">A27</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="firebrick">::  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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_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="E1:39_5_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="#E1:39_5"><span class="lab"><font color="Green" title="E48">A27</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_1_1_1"><span class="kw">case </span></a><a NAME="E1:39_5_1_1_1"/><span class="lab"><font color="Green" title="E49">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  <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="firebrick">::  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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E50">A29</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_5_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A30</font></span>: 
(  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  or  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_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="E1:39_5_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="#E1:39_5_1_1_1_1"><span class="lab"><font color="Green" title="E51">A30</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_1_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:39_5_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A31</font></span>: 
( (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp;  not  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:39_5_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A32</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="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</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="39_5_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="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="E1:39_5_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="c12">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="c12">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="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E1:39_5_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a></span>;<br/>
<a NAME="E2:39_5_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a></span>)</span> <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#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_5_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:39_5_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E54">A33</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#K13" title="BVFUNC_1:func.13">'imp'</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="39_5_1_1_1_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c12">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="E1:39_5_1_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c12">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="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E1:39_5_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a></span>;<br/>
<a NAME="E2:39_5_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a></span>)</span> <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#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:39_5_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="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="#E1:39_5_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><a NAME="E5:39_5_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A34</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="#E47:39"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><a NAME="E6:39_5_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A35</font></span>: 
 <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="#E1:39_5_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><a NAME="E7:39_5_1_1_1_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#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="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E8:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:39"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E3:39_5_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="txt" href="#E2:39_5_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="txt" href="#E6:39_5_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="txt" href="#E5:39_5_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A34</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_1_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:39_5_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E52">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#K7" title="MARGREL1:func.7">FALSE</a>  ) &amp;  not  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:39_5_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E53">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  <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="39_5_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="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c12">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="E1:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></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="c12">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="c12">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="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="bvfunc_1.html#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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="c12">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="#E1:39_5_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A36</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E54">A38</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1_1_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_5_1_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E55">A39</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  holds contradiction</span><br/><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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="c12">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="#E1:39_5_1_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E55">A39</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_5_1_1_1_1_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_5_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:39_5_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E55">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="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</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="39_5_1_1_1_1_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="E1:39_5_1_1_1_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="c12">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="c12">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="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E1:39_5_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A36</font></span></a></span>;<br/>
<a NAME="E2:39_5_1_1_1_1_1_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a></span>)</span> <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#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_5_1_1_1_1_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:39_5_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A41</font></span>: 
 <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="#E1:39_5_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><a NAME="E6:39_5_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="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="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><a NAME="E7:39_5_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A42</font></span>: 
<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#K13" title="BVFUNC_1:func.13">'imp'</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="#E24:39"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E5:39_5_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="ref" href="bvfunc_1.html#T22" target="_self" title="BVFUNC_1:th.22">Th25</a></span>;<br/><a NAME="E8:39_5_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A43</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="#E1:39_5_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><a NAME="E9:39_5_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="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><span class="kw">hence </span><a NAME="E10:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:39"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E35:39"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E3:39_5_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="txt" href="#E4:39_5_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:39_5_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E7:39_5_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_1_1_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:39_5_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E52">A44</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="firebrick">::  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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:39_5_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E53">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#K13" title="BVFUNC_1:func.13">'imp'</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="39_5_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="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c12">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="E1:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></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="c12">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="c12">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="#E1:39_5_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="ref" href="bvfunc_1.html#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:39_5_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E54">A46</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>  <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="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><a NAME="E4:39_5_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E55">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="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</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="39_5_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="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="E1:39_5_1_1_1_1_1_1_3_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c12">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="c12">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="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="bvfunc_1.html#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:39_5_1_1_1_1_1_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:39_5_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="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#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="#E1:39_5_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E6:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:39"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E12:39"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E24:39"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E2:39_5_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="txt" href="#E4:39_5_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="txt" href="#E3:39_5_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:39_5_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A48</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="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</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="39_5_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="E1:39_5_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c12">x</font></span>)</span> <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#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D19" title="MARGREL1:def.19">MARGREL1:def 19</a></span>;<br/>
<a NAME="E2:39_5_1_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c12">x</font></span>)</span> <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#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D4" target="_self" title="BVFUNC_1:def.4">Def7</a></span>;<br/>
<a NAME="E3:39_5_1_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c12">x</font></span>)</span> <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#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <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></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/>
<a NAME="E4:39_5_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A49</font></span>: 
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c12">x</font></span>)</span> <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="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T10" target="_self" title="BVFUNC_1:th.10">Th13</a></span>;<br/>
<a NAME="E5:39_5_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c12">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:39_5_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="#E4:39_5_1_1_1_2_1"><span class="lab"><font color="Green" title="E52">A49</font></span></a>, <a class="ref" href="bvfunc_1.html#D11" target="_self" title="BVFUNC_1:def.11">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E2:39_5_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A50</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>  <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="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:39_5_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A51</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="E4:39_5_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A52</font></span>: 
 not  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><a NAME="E5:39_5_1_1_1_2"/><span class="lab"><font color="Green" title="E55">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#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="#E3:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E53">A51</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><a NAME="E6:39_5_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#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="#E4:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E54">A52</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><a NAME="E7:39_5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A54</font></span>: 
<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#K13" title="BVFUNC_1:func.13">'imp'</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="#E11:39"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E2:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E52">A50</font></span></a>, <a class="ref" href="bvfunc_1.html#T22" target="_self" title="BVFUNC_1:th.22">Th25</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">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="E9:39_5_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A55</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k4</font>
 <span class="kw">and </span><br/><a NAME="E10:39_5_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A56</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">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="E11:39_5_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">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="c13">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="E13:39_5_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A57</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">k3</font>
 <span class="kw">and </span><br/><a NAME="E14:39_5_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A58</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">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="E15:39_5_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">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="E16:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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="39_5_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c14">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="c14">x</font></span><br/>

<a NAME="E1:39_5_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c14">x</font></span>)</span></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="c14">x</font></span>)</span> <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#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">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="c14">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D19" title="MARGREL1:def.19">MARGREL1:def 19</a></span>;<br/>
<a NAME="E2:39_5_1_1_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c14">x</font></span>)</span></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="c14">x</font></span>)</span> <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#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">d</font></span>)</span> <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="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D4" target="_self" title="BVFUNC_1:def.4">Def7</a></span>;<br/>
<a NAME="E3:39_5_1_1_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c14">x</font></span>)</span></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="c14">x</font></span>)</span> <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#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K16" title="BVFUNC_1:func.16">I_el</a> <font color="Maroon" title="c1">Y</font></span>)</span></span>)</span> <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="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E1:39_5_1_1_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/>
<a NAME="E4:39_5_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A59</font></span>: 
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c14">x</font></span>)</span></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="c14">x</font></span>)</span> <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#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="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T2" target="_self" title="BVFUNC_1:th.2">Th5</a></span>;<br/>
<a NAME="E5:39_5_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c14">x</font></span>)</span></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="c14">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:39_5_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c14">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="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_5_1_1_1_2_2"><span class="lab"><font color="Green" title="E61">A59</font></span></a>, <a class="ref" href="bvfunc_1.html#T9" target="_self" title="BVFUNC_1:th.9">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E17:39_5_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="c13">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="c12">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="#E13:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E59">A57</font></span></a>, <a class="txt" href="#E9:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E57">A55</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:39"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E35:39"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E2:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E52">A50</font></span></a>, <a class="txt" href="#E5:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E55">A53</font></span></a>, <a class="txt" href="#E13:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E59">A57</font></span></a>, <a class="txt" href="#E14:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E60">A58</font></span></a>, <a class="txt" href="#E9:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E57">A55</font></span></a>, <a class="txt" href="#E10:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E58">A56</font></span></a>, <a class="txt" href="#E1:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E51">A48</font></span></a>, <a class="txt" href="#E7:39_5_1_1_1_2"><span class="lab"><font color="Green" title="E56">A54</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_5_1_1_1"><span class="lab"><font color="Green" title="E50">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_1_1_2"><span class="kw">case </span></a><a NAME="E1:39_5_1_1_2"/><span class="lab"><font color="Green" title="E49">A60</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="firebrick">::  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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E50">A61</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_5_1_1_2_1"/><span class="lab"><font color="Green" title="E51">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="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  or  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_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="E1:39_5_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="#E1:39_5_1_1_2_1"><span class="lab"><font color="Green" title="E51">A62</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_1_1_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:39_5_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A63</font></span>: 
( (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp;  not  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:39_5_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A64</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="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</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="39_5_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="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="E1:39_5_1_1_2_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="c12">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="c12">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="#E1:39_5_1_1_2"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="txt" href="#E1:39_5_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E52">A63</font></span></a></span>;<br/>
<a NAME="E2:39_5_1_1_2_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a></span>)</span> <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#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_5_1_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E54">A65</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1_2_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_5_1_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E55">A66</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="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  holds contradiction</span><br/><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:39_5_1_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="#E1:39_5_1_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A66</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_5_1_1_2_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_5_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E53">A64</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:39_5_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E55">A67</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#K13" title="BVFUNC_1:func.13">'imp'</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="39_5_1_1_2_1_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c12">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="E1:39_5_1_1_2_1_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="c12">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="c12">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="#E1:39_5_1_1_2"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="txt" href="#E1:39_5_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E52">A63</font></span></a></span>;<br/>
<a NAME="E2:39_5_1_1_2_1_1_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a></span>)</span> <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#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_5_1_1_2_1_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:39_5_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A68</font></span>: 
 <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="#E1:39_5_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E52">A63</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><a NAME="E6:39_5_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="#E1:39_5_1_1_2"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><a NAME="E7:39_5_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A69</font></span>: 
<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#K13" title="BVFUNC_1:func.13">'imp'</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="#E35:39"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E5:39_5_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E56">A68</font></span></a>, <a class="ref" href="bvfunc_1.html#T22" target="_self" title="BVFUNC_1:th.22">Th25</a></span>;<br/><a NAME="E8:39_5_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A70</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="#E1:39_5_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E52">A63</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><a NAME="E9:39_5_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="#E1:39_5_1_1_2"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><span class="kw">hence </span><a NAME="E10:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:39"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E24:39"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E4:39_5_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E55">A67</font></span></a>, <a class="txt" href="#E3:39_5_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E54">A65</font></span></a>, <a class="txt" href="#E8:39_5_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E58">A70</font></span></a>, <a class="txt" href="#E7:39_5_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E57">A69</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_1_1_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:39_5_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A71</font></span>: 
( (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  ) &amp;  not  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:39_5_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A72</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#K13" title="BVFUNC_1:func.13">'imp'</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="39_5_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="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c12">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="E1:39_5_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="c12">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="c12">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="#E1:39_5_1_1_2"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="txt" href="#E1:39_5_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E52">A71</font></span></a></span>;<br/>
<a NAME="E2:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a></span>)</span> <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#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:39_5_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A73</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>  <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="#E1:39_5_1_1_2"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><a NAME="E4:39_5_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E55">A74</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="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</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="39_5_1_1_2_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="E1:39_5_1_1_2_1_1_1_2_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c12">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="c12">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="#E1:39_5_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E52">A71</font></span></a>, <a class="ref" href="bvfunc_1.html#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:39_5_1_1_2_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:39_5_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="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> &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#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="#E1:39_5_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E52">A71</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><span class="kw">hence </span><a NAME="E6:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:39"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E12:39"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E46:39"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="txt" href="#E2:39_5_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E53">A72</font></span></a>, <a class="txt" href="#E4:39_5_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E55">A74</font></span></a>, <a class="txt" href="#E3:39_5_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E54">A73</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_1_1_2_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:39_5_1_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E52">A75</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="firebrick">::  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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><a NAME="E2:39_5_1_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E53">A76</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="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</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="39_5_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="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="E1:39_5_1_1_2_1_1_1_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c12">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="c12">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="#E1:39_5_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E52">A75</font></span></a>, <a class="ref" href="bvfunc_1.html#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:39_5_1_1_2_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:39_5_1_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E54">A77</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#K13" title="BVFUNC_1:func.13">'imp'</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="39_5_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="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c12">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="E1:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></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="c12">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="c12">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="#E1:39_5_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E52">A75</font></span></a>, <a class="ref" href="bvfunc_1.html#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:39_5_1_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E55">A78</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>  <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="#E1:39_5_1_1_2"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><a NAME="E5:39_5_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="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="#E1:39_5_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E52">A75</font></span></a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><a NAME="E6:39_5_1_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A79</font></span>: 
<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#K13" title="BVFUNC_1:func.13">'imp'</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="#E46:39"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="txt" href="#E4:39_5_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A78</font></span></a>, <a class="ref" href="bvfunc_1.html#T22" target="_self" title="BVFUNC_1:th.22">Th25</a></span>;<br/><a NAME="E7:39_5_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="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="#E1:39_5_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E52">A75</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><span class="kw">hence </span><a NAME="E8:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:39"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E24:39"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E3:39_5_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E54">A77</font></span></a>, <a class="txt" href="#E2:39_5_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E53">A76</font></span></a>, <a class="txt" href="#E4:39_5_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A78</font></span></a>, <a class="txt" href="#E6:39_5_1_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E56">A79</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1_2_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:39_5_1_1_2_2"/><span class="lab"><font color="Green" title="E51">A80</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>  <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="#E1:39_5_1_1_2"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><a NAME="E2:39_5_1_1_2_2"/><span class="lab"><font color="Green" title="E52">A81</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#K13" title="BVFUNC_1:func.13">'imp'</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="39_5_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c12">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="E1:39_5_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></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="c12">x</font></span>)</span> <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#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c12">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D19" title="MARGREL1:def.19">MARGREL1:def 19</a></span>;<br/>
<a NAME="E2:39_5_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></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="c12">x</font></span>)</span> <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#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p3">(<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>)</span> <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="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E1:39_5_1_1_2_2"><span class="lab"><font color="Green" title="E51">A80</font></span></a>, <a class="ref" href="bvfunc_1.html#D4" target="_self" title="BVFUNC_1:def.4">Def7</a></span>;<br/>
<a NAME="E3:39_5_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></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="c12">x</font></span>)</span> <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="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="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T2" target="_self" title="BVFUNC_1:th.2">Th5</a></span>;<br/>
<a NAME="E4:39_5_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></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="c12">x</font></span>)</span> <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="c12">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#T10" target="_self" title="BVFUNC_1:th.10">Th13</a></span>;<br/>
<a NAME="E5:39_5_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c12">x</font></span>)</span></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="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D11" target="_self" title="BVFUNC_1:def.11">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:39_5_1_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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="c12">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#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c12">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="E4:39_5_1_1_2_2"/><span class="lab"><font color="Green" title="E53">A82</font></span>: 
 <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k4</font>
 <span class="kw">and </span><br/><a NAME="E5:39_5_1_1_2_2"/><span class="lab"><font color="Green" title="E54">A83</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">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="E6:39_5_1_1_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">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="c13">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="E8:39_5_1_1_2_2"/><span class="lab"><font color="Green" title="E55">A84</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">k3</font>
 <span class="kw">and </span><br/><a NAME="E9:39_5_1_1_2_2"/><span class="lab"><font color="Green" title="E56">A85</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">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="E10:39_5_1_1_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">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="E11:39_5_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="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</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> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <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="c14">x</font></span><br/>

<a NAME="E1:39_5_1_1_2_2_2"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c14">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c14">x</font></span>)</span> <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#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">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="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#D19" title="MARGREL1:def.19">MARGREL1:def 19</a></span>;<br/>
<a NAME="E2:39_5_1_1_2_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c14">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c14">x</font></span>)</span> <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#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="bvfunc_1.html#K7" title="BVFUNC_1:func.7">'or'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D4" target="_self" title="BVFUNC_1:def.4">Def7</a></span>;<br/>
<a NAME="E3:39_5_1_1_2_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<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="c14">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5" title="XBOOLEAN:func.5">'or'</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="c14">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <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="c14">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E1:39_5_1_1_2_2"><span class="lab"><font color="Green" title="E51">A80</font></span></a>, <a class="ref" href="bvfunc_1.html#T9" target="_self" title="BVFUNC_1:th.9">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:39_5_1_1_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <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="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bvfunc_1.html#D8" target="_self" title="BVFUNC_1:def.8">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E12:39_5_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="c13">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="c12">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="#E8:39_5_1_1_2_2"><span class="lab"><font color="Green" title="E55">A84</font></span></a>, <a class="txt" href="#E4:39_5_1_1_2_2"><span class="lab"><font color="Green" title="E53">A82</font></span></a></span>;<br/><a NAME="E13:39_5_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K1" title="BVFUNC_1:func.1">'not'</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_5_1_1_2_2"><span class="lab"><font color="Green" title="E55">A84</font></span></a>, <a class="txt" href="#E9:39_5_1_1_2_2"><span class="lab"><font color="Green" title="E56">A85</font></span></a>, <a class="txt" href="#E4:39_5_1_1_2_2"><span class="lab"><font color="Green" title="E53">A82</font></span></a>, <a class="txt" href="#E5:39_5_1_1_2_2"><span class="lab"><font color="Green" title="E54">A83</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/><a NAME="E14:39_5_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A86</font></span>: 
 <a href="bvfunc_1.html#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bvfunc_1.html#K1" title="BVFUNC_1:func.1">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#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#T19" target="_self" title="BVFUNC_1:th.19">Th22</a></span>;<br/><span class="kw">assume </span><a NAME="E15:39_5_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#K8" title="MARGREL1:func.8">TRUE</a>  &amp;  not  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds  <font color="Maroon" title="c2">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><a NAME="E16:39_5_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A87</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> &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="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a>, <a class="ref" href="bvfunc_1.html#D14" target="_self" title="BVFUNC_1:def.14">Def17</a></span>;<br/><a NAME="E17:39_5_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="#E1:39_5_1_1_2"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a></span>;<br/><span class="kw">hence </span><a NAME="E18:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:39"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E12:39"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E35:39"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E46:39"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="txt" href="#E2:39_5_1_1_2_2"><span class="lab"><font color="Green" title="E52">A81</font></span></a>, <a class="txt" href="#E14:39_5_1_1_2_2"><span class="lab"><font color="Green" title="E57">A86</font></span></a>, <a class="txt" href="#E16:39_5_1_1_2_2"><span class="lab"><font color="Green" title="E58">A87</font></span></a>, <a class="ref" href="bvfunc_1.html#D13" target="_self" title="BVFUNC_1:def.13">Def16</a>, <a class="ref" href="bvfunc_1.html#T2" target="_self" title="BVFUNC_1:th.2">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_5_1_1_2"><span class="lab"><font color="Green" title="E50">A61</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_5_1_1_3"><span class="kw">case </span></a><a NAME="E1:39_5_1_1_3"/><span class="lab"><font color="Green" title="E49">A88</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> holds <br/> (  <a href="bvfunc_1.html#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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="bvfunc_1.html#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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><a NAME="E2:39_5_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="c12">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="#E1:39_5_1_1_3"><span class="lab"><font color="Green" title="E49">A88</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_5_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_5_1_1_3"><span class="lab"><font color="Green" title="E49">A88</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:39_5"/>
(  <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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_6"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39_6"/><span class="lab"><font color="Green" title="E48">A89</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="E2:39_6"/><span class="lab"><font color="Green" title="E49">A90</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="firebrick">::  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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_6_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_6_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="E1:39_6_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#T21" target="_self" title="BVFUNC_1:th.21">Th24</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_6_1_1_1"><span class="kw">case </span></a><a NAME="E1:39_6_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="firebrick">::  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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:39_6_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_6"><span class="lab"><font color="Green" title="E49">A90</font></span></a>, <a class="ref" href="bvfunc_1.html#D10" target="_self" title="BVFUNC_1:def.10">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_6_1_1_2"><span class="kw">case </span></a><a NAME="E1:39_6_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="firebrick">::  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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:39_6_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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_6"><span class="lab"><font color="Green" title="E48">A89</font></span></a>, <a class="ref" href="bvfunc_1.html#D11" target="_self" title="BVFUNC_1:def.11">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:39_6"/>
(  <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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E51:39"/>
(  <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#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</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#K17" title="BVFUNC_1:func.17">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bvfunc_1.html#K13" title="BVFUNC_1:func.13">'imp'</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#K13" title="BVFUNC_1:func.13">'imp'</a> <font color="Maroon" title="c3">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:39"><span class="lab"><font color="Green" title="E47">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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