<?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/>



<a NAME="E1:35"/><i><font color="Green">E19</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>  ) iff 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"><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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_1"/><i><font color="Green">E20</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>  ) implies 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"><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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_1"/><i><font color="Green">E29</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> 
 ;<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="E2:35_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/>;<br/>
<a NAME="E3:35_1_1"/><b>then </b>
 <a href="margrel1.html#K9">'not'</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#K7">FALSE</a> 
 
;<br/>
<b>hence </b><a NAME="E4:35_1_1"/>
<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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:35_1"/>
( ( 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"><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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) implies 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>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_2"/><i><font color="Green">E31</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"><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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</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="E2:35_1_2"/>
<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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:35_1_2"/><b>then </b>
 <a href="margrel1.html#K9">'not'</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#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:35_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> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:35_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>  ) iff 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"><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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:35"/><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  <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>  ) iff 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"><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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<a NAME="E1:35_2"/><i><font color="Green">E44</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>  ) implies 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"><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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_2_1"/><i><font color="Green">E45</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> 
 ;<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="E2:35_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/>;<br/>
<a NAME="E3:35_2_1"/><b>then </b>
 <a href="margrel1.html#K9">'not'</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> 
 
;<br/>
<b>hence </b><a NAME="E4:35_2_1"/>
<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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:35_2"/>
( ( 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"><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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) implies 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>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_2_2"/><i><font color="Green">E46</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"><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> <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="E2:35_2_2"/>
<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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:35_2_2"/><b>then </b>
 <a href="margrel1.html#K9">'not'</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><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:35_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> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:35_2"/>
( ( 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>  ) iff 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"><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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_3"/><i><font color="Green">E49</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="35_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_3_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#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">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>  ) ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_1"><b>case </b></a><a NAME="E1:35_3_1_1_1"/><i><font color="Green">E50</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="E2:35_3_1_1_1"/><b>then </b>
 <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="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:35_3_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="valuat_1.html#K5">'not'</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#K16" target="_self">O_el</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><a NAME="E4:35_3_1_1_1"/><i><font color="Green">E52</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"><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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_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="E1:35_3_1_1_1_1"/>
 <a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a>  <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
;<br/>
<a NAME="E2:35_3_1_1_1_1"/><b>then </b>
 <a href="margrel1.html#K9">'not'</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#K7">FALSE</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:35_3_1_1_1_1"/>
<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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:35_3_1_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="bvfunc_1.html#K18" target="_self">B_INF</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#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#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:35_3_1_1_1"/>
 <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> <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#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="bvfunc_1.html#E1:35"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E7:35_3_1_1_1"/>
(  <a href="valuat_1.html#K5">'not'</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#K19" target="_self">B_SUP</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> &amp;  <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> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, , , <a class="txt" href="bvfunc_1.html#E1:35"><i><font color="Green">E19</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_2"><b>case </b></a><a NAME="E1:35_3_1_1_2"/><i><font color="Green">E57</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="E2:35_3_1_1_2"/><b>then </b>
 <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> <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#K16" target="_self">O_el</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:35"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E3:35_3_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
 <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> <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/>;<br/><a NAME="E4:35_3_1_1_2"/><i><font color="Green">E76</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"><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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_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="E1:35_3_1_1_2_1"/>
 <a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a>  <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
;<br/>
<a NAME="E2:35_3_1_1_2_1"/><b>then </b>
 <a href="margrel1.html#K9">'not'</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="E3:35_3_1_1_2_1"/>
<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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:35_3_1_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</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#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#E1:35"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E6:35_3_1_1_2"/>
 <a href="valuat_1.html#K5">'not'</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="valuat_1.html#K5">'not'</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>
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E7:35_3_1_1_2"/>
(  <a href="valuat_1.html#K5">'not'</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#K19" target="_self">B_SUP</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> &amp;  <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> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, , , <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="35_3_1_1_3"><b>case </b></a><a NAME="E1:35_3_1_1_3"/><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  <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; ( 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"><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="E2:35_3_1_1_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#K8">TRUE</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:35_3_1_1_3"/>
(  <a href="valuat_1.html#K5">'not'</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#K19" target="_self">B_SUP</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> &amp;  <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> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:35_3"/>
(  <a href="valuat_1.html#K5">'not'</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#K19" target="_self">B_SUP</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> &amp;  <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> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_4"/><i><font color="Green">E82</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="E2:35_4"/><b>then </b>
 <a href="valuat_1.html#K5">'not'</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="valuat_1.html#K5">'not'</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>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:35_4"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="valuat_1.html#K5">'not'</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/>;<br/><a NAME="E4:35_4"/><i><font color="Green">E84</font></i>: 
 <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> <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#K17" target="_self">I_el</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="bvfunc_1.html#E1:35"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E5:35_4"/>
 <a href="bvfunc_1.html#K18" target="_self">B_INF</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#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#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E6:35_4"/>
(  <a href="valuat_1.html#K5">'not'</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#K19" target="_self">B_SUP</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> &amp;  <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> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, , , , <a class="txt" href="bvfunc_1.html#E1:35"><i><font color="Green">E19</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:35"/>
(  <a href="valuat_1.html#K5">'not'</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#K19" target="_self">B_SUP</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> &amp;  <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> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
