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

<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">d</font> be  <a href="funct_1.html#V3">constant</a>  <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>





<a NAME="E1:43"/><i><font color="Green">E19</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:43"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:43"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:43"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_1"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:43_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
;<br/>
<b>hence </b><a NAME="E4:43_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">k3</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:43"/><i><font color="Green">E44</font></i>: 
( <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k3</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k3</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">k4</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:43"/><i><font color="Green">E45</font></i>: 
(  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k4</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E10:43"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">k3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font>
 
<b>by </b><i>, , <a class="txt" href="bvfunc_1.html#E3:43"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E11:43"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="txt" href="bvfunc_1.html#E3:43"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E12:43"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_2"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_2"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:43_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">k3</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E14:43"/><i><font color="Green">E50</font></i>: 
( <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k3</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k3</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">k4</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E16:43"/><i><font color="Green">E51</font></i>: 
(  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k4</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E17:43"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">k3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font>
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E4:43"><i><font color="Green">E31</font></i></a>, , </i>;<br/>
<a NAME="E18:43"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E19:43"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_3"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_3"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:43_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:43_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K7">FALSE</a>  <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:43_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">k3</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E21:43"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k3</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k3</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">k4</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E23:43"/><i><font color="Green">E76</font></i>: 
(  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k4</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E24:43"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">k3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E25:43"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E26:43"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_4"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:43_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K8">TRUE</a>  <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
;<br/>
<a NAME="E4:43_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
;<br/>
<b>hence </b><a NAME="E5:43_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">k3</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E28:43"/><i><font color="Green">E82</font></i>: 
( <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k3</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k3</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">k4</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E30:43"/><i><font color="Green">E83</font></i>: 
(  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k4</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E31:43"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">k3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E32:43"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<div><i><font color="Green">E92</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_5"/><i><font color="Green">E93</font></i>: 
( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_5_1_1"/>
( ( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) or ( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) or ( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) ) )
 </span><b>by </b><i><a class="txt" href="bvfunc_1.html#E7:43"><i><font color="Green">E44</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1_1"><b>case </b></a><a NAME="E1:43_5_1_1_1"/><i><font color="Green">E94</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div class="add"><div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_5_1_1_1_1"/><i><font color="Green">E96</font></i>: 
( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_5_1_1_1_1_1_1"/>
( ( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) or ( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) or ( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) ) )
 </span><b>by </b><i><a class="ref" href="bvfunc_1.html#D6" target="_self">Def6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:43_5_1_1_1_1_1_1_1"/><i><font color="Green">E97</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div class="add"><a NAME="E2:43_5_1_1_1_1_1_1_1"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_1_1_1_1_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E2:43_5_1_1_1_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:43_5_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:43_5_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E3:43_5_1_1_1_1_1_1_1"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_1_1_1_1_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E2:43_5_1_1_1_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:43_5_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:43_5_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E4:43_5_1_1_1_1_1_1_1"/><i><font color="Green">E102</font></i>: 
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E5:43_5_1_1_1_1_1_1_1"/><i><font color="Green">E103</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:43_5_1_1_1_1_1_1_1"/><i><font color="Green">E104</font></i>: 
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E7:43_5_1_1_1_1_1_1_1"/>
<font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E8:43_5_1_1_1_1_1_1_1"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , , <a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bvfunc_1.html#E19:43"><i><font color="Green">E53</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:43_5_1_1_1_1_1_1_2"/><i><font color="Green">E105</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div class="add"><a NAME="E2:43_5_1_1_1_1_1_1_2"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_1_1_1_1_2_1"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_5_1_1_1_1_1_1_2_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E3:43_5_1_1_1_1_1_1_2_1"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E4:43_5_1_1_1_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K7">FALSE</a>  <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:43_5_1_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E3:43_5_1_1_1_1_1_1_2"/><i><font color="Green">E109</font></i>: 
not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_1_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_5_1_1_1_1_1_1_2_2_1"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/><a NAME="E2:43_5_1_1_1_1_1_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a></i>;<br/><b>hence </b><a NAME="E3:43_5_1_1_1_1_1_1_2_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_5_1_1_1_1_1_1_2_2"/>
not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E4:43_5_1_1_1_1_1_1_2"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_1_1_1_1_2_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E2:43_5_1_1_1_1_1_1_2_3"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:43_5_1_1_1_1_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:43_5_1_1_1_1_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E5:43_5_1_1_1_1_1_1_2"/><i><font color="Green">E113</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:43_5_1_1_1_1_1_1_2"/><i><font color="Green">E114</font></i>: 
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E7:43_5_1_1_1_1_1_1_2"/><i><font color="Green">E115</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E8:43_5_1_1_1_1_1_1_2"/>
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E9:43_5_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="txt" href="bvfunc_1.html#E19:43"><i><font color="Green">E53</font></i></a>, </i>;<br/><b>hence </b><a NAME="E10:43_5_1_1_1_1_1_1_2"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#D5" target="_self">Def5</a>, <a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a>, , <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:43_5_1_1_1_1_1_1_3"/><i><font color="Green">E116</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 ;<br/><div class="add"><a NAME="E2:43_5_1_1_1_1_1_1_3"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_1_1_1_1_3_1"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_5_1_1_1_1_1_1_3_1"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:43_5_1_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:43_5_1_1_1_1_1_1_3"/><i><font color="Green">E119</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_1_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_1_1_1_1_3_2"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_5_1_1_1_1_1_1_3_2"/>
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:43_5_1_1_1_1_1_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:43_5_1_1_1_1_1_1_3"/><i><font color="Green">E121</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E5:43_5_1_1_1_1_1_1_3"/><i><font color="Green">E122</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:43_5_1_1_1_1_1_1_3"/>
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E7:43_5_1_1_1_1_1_1_3"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , , , , <a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:43_5_1_1_1_1"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_5_1_1_1_2"/><i><font color="Green">E123</font></i>: 
( not <!--HUMANRECFAILED-->for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; not <!--HUMANRECFAILED-->for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><a NAME="E2:43_5_1_1_1_2"/><i><font color="Green">E124</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:43_5_1_1_1_2"/><i><font color="Green">E125</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D6" target="_self">Def6</a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E4:43_5_1_1_1_2"/><i><font color="Green">E126</font></i>: 
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D6" target="_self">Def6</a>, <a class="txt" href="bvfunc_1.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E5:43_5_1_1_1_2"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_1_2_1"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_5_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>
<a NAME="E3:43_5_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">d</font></span>)</span> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:43_5_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:43_5_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:43_5_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a>, </i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">k3</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:43_5_1_1_1_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k3</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k3</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">k4</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:43_5_1_1_1_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k4</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E10:43_5_1_1_1_2"/><i><font color="Green">E131</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">k3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bvfunc_1.html#E19:43"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E11:43_5_1_1_1_2"/><i><font color="Green">E132</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_1_2_2"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_5_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>
<a NAME="E3:43_5_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:43_5_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:43_5_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:43_5_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E12:43_5_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a>, </i>;<br/><b>hence </b><a NAME="E13:43_5_1_1_1_2"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#D5" target="_self">Def5</a>, <a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, , <a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bvfunc_1.html#E19:43"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:43_5_1_1_1"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E11:43"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1_2"><b>case </b></a><a NAME="E1:43_5_1_1_2"/><i><font color="Green">E134</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div class="add"><div><i><font color="Green">E135</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_5_1_1_2_1"/><i><font color="Green">E136</font></i>: 
( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_5_1_1_2_1_1_1"/>
( ( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) or ( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) or ( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) ) )
 </span><b>by </b><i><a class="ref" href="bvfunc_1.html#D6" target="_self">Def6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:43_5_1_1_2_1_1_1_1"/><i><font color="Green">E137</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div class="add"><a NAME="E2:43_5_1_1_2_1_1_1_1"/><i><font color="Green">E138</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_2_1_1_1_1_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E2:43_5_1_1_2_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:43_5_1_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:43_5_1_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E3:43_5_1_1_2_1_1_1_1"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_2_1_1_1_1_2"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E2:43_5_1_1_2_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:43_5_1_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, </i>;<br/>
<a NAME="E4:43_5_1_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K7">FALSE</a>  <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
;<br/>
<b>hence </b><a NAME="E5:43_5_1_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E4:43_5_1_1_2_1_1_1_1"/><i><font color="Green">E142</font></i>: 
not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_1_1_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_5_1_1_2_1_1_1_1_3_1"/><i><font color="Green">E143</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/><a NAME="E2:43_5_1_1_2_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a></i>;<br/><b>hence </b><a NAME="E3:43_5_1_1_2_1_1_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_5_1_1_2_1_1_1_1_3"/>
not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E5:43_5_1_1_2_1_1_1_1"/><i><font color="Green">E144</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:43_5_1_1_2_1_1_1_1"/><i><font color="Green">E145</font></i>: 
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E7:43_5_1_1_2_1_1_1_1"/><i><font color="Green">E146</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E8:43_5_1_1_2_1_1_1_1"/>
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E9:43_5_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D5" target="_self">Def5</a>, <a class="txt" href="bvfunc_1.html#E19:43"><i><font color="Green">E53</font></i></a>, </i>;<br/><b>hence </b><a NAME="E10:43_5_1_1_2_1_1_1_1"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , , <a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a>, , <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:43_5_1_1_2_1_1_1_2"/><i><font color="Green">E147</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; not for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div class="add"><a NAME="E2:43_5_1_1_2_1_1_1_2"/><i><font color="Green">E148</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_2_1_1_1_2_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E2:43_5_1_1_2_1_1_1_2_1"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:43_5_1_1_2_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<a NAME="E4:43_5_1_1_2_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K8">TRUE</a>  <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
;<br/>
<b>hence </b><a NAME="E5:43_5_1_1_2_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E3:43_5_1_1_2_1_1_1_2"/><i><font color="Green">E150</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_2_1_1_1_2_2"/><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_5_1_1_2_1_1_1_2_2"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:43_5_1_1_2_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K8">TRUE</a>  <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:43_5_1_1_2_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E4:43_5_1_1_2_1_1_1_2"/><i><font color="Green">E152</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E5:43_5_1_1_2_1_1_1_2"/><i><font color="Green">E153</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:43_5_1_1_2_1_1_1_2"/>
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E7:43_5_1_1_2_1_1_1_2"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , , , <a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:43_5_1_1_2_1_1_1_3"/><i><font color="Green">E154</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 ;<br/><div class="add"><a NAME="E2:43_5_1_1_2_1_1_1_3"/><i><font color="Green">E155</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_2_1_1_1_3_1"/><i><font color="Green">E156</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_5_1_1_2_1_1_1_3_1"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:43_5_1_1_2_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:43_5_1_1_2_1_1_1_3"/><i><font color="Green">E157</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_1_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_2_1_1_1_3_2"/><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_5_1_1_2_1_1_1_3_2"/>
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:43_5_1_1_2_1_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="margrel1.html#K8">TRUE</a>  <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:43_5_1_1_2_1_1_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E4:43_5_1_1_2_1_1_1_3"/><i><font color="Green">E159</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E5:43_5_1_1_2_1_1_1_3"/><i><font color="Green">E160</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:43_5_1_1_2_1_1_1_3"/>
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E7:43_5_1_1_2_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:43_5_1_1_2_1_1_1_3"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , , <a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:43_5_1_1_2_1"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_5_1_1_2_2"/><i><font color="Green">E161</font></i>: 
( not <!--HUMANRECFAILED-->for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; not <!--HUMANRECFAILED-->for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><a NAME="E2:43_5_1_1_2_2"/><i><font color="Green">E162</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:43_5_1_1_2_2"/><i><font color="Green">E163</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>
<a NAME="E2:43_5_1_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, </i>;<br/>
<a NAME="E3:43_5_1_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:43_5_1_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:43_5_1_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:43_5_1_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:43_5_1_1_2_2"/><i><font color="Green">E164</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E1:43_5_1_1_2_2_2"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>
<a NAME="E2:43_5_1_1_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:43_5_1_1_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="txt" href="bvfunc_1.html#E12:43"><i><font color="Green">E48</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:43_5_1_1_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">k3</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:43_5_1_1_2_2"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k3</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k3</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">k4</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:43_5_1_1_2_2"/><i><font color="Green">E166</font></i>: 
(  <a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">k4</font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E9:43_5_1_1_2_2"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">k3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k4</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E14:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E10:43_5_1_1_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E16:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bvfunc_1.html#E18:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E11:43_5_1_1_2_2"/><b>then </b><i><font color="Green">E167</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="valuat_1.html#K5">'not'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:43"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E12:43_5_1_1_2_2"/><i><font color="Green">E168</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E13:43_5_1_1_2_2"/><i><font color="Green">E169</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D6" target="_self">Def6</a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E14:43_5_1_1_2_2"/>
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D6" target="_self">Def6</a>, <a class="txt" href="bvfunc_1.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E15:43_5_1_1_2_2"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , <a class="ref" href="bvfunc_1.html#D5" target="_self">Def5</a>, , , <a class="txt" href="bvfunc_1.html#E19:43"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:43_5_1_1_2"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E11:43"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_1_3"><b>case </b></a><a NAME="E1:43_5_1_1_3"/><i><font color="Green">E170</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) )
 ;<br/><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/><a NAME="E2:43_5_1_1_3"/>
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E3:43_5_1_1_3"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:43"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:43_5"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_6"/><i><font color="Green">E171</font></i>: 
( not <!--HUMANRECFAILED-->for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; not <!--HUMANRECFAILED-->for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_6_1_1"/>
( <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font> or <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_1_1_1"><b>case </b></a><a NAME="E1:43_6_1_1_1"/>
<font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_6_1_1_1"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E7:43"><i><font color="Green">E44</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_1_1_2"><b>case </b></a><a NAME="E1:43_6_1_1_2"/>
<font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17" target="_self">I_el</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_6_1_1_2"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E7:43"><i><font color="Green">E44</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:43_6"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E35:43"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">d</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K12" target="_self">'imp'</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E5:43"><i><font color="Green">E43</font></i></a></i>;<br/>


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