<?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> , <font color="Maroon">b</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/>



<div><i><font color="Green">E19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_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  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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/><a NAME="E2:42_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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_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="E1:42_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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#D6">VALUAT_1:def 6</a></i>;<br/>
<a NAME="E2:42_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<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="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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>thus </b><a NAME="E3:42_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="ref" href="xboolean.html#T101">XBOOLEAN:101</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:42_1"/><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">b</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="42_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="E1:42_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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="E2:42_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<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="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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#D6">VALUAT_1:def 6</a></i>;<br/>
<b>thus </b><a NAME="E3:42_1_2"/>
<font color="Maroon">b</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="ref" href="xboolean.html#T101">XBOOLEAN:101</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E45</font></i>: <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="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">b</font></span>)</span> = 
<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="valuat_1.html#K6">'&amp;'</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="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<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="valuat_1.html#K6">'&amp;'</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="ref" href="bvfunc_1.html#D3" target="_self">Def3</a></i>
;<br/><a NAME="E5:42_1"/><i><font color="Green">E46</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#K6" target="_self">'or'</a> <font color="Maroon">b</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="42_1_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_4_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"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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="E2:42_1_4_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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"><font color="Maroon">a</font> <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">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:42_1_4_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="E4:42_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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#E2:42_1"><i><font color="Green">E29</font></i></a></i>;<br/><b>hence </b><a NAME="E5:42_1_4_1"/>
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#K6" target="_self">'or'</a> <font color="Maroon">b</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:42_1_4"/>
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#K6" target="_self">'or'</a> <font color="Maroon">b</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/>


</div><b>end;</b></div><a NAME="E6:42_1"/><i><font color="Green">E49</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  <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="42_1_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_5_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#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="E2:42_1_5_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/><b>hence </b><a NAME="E3:42_1_5_1"/>
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><b>end;</b></div>
<b>hence </b><a NAME="E2:42_1_5"/>
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><b>end;</b></div><a NAME="E7:42_1"/><i><font color="Green">E50</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  <font color="Maroon">b</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="42_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_6_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">b</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="E2:42_1_6_1"/>
<font color="Maroon">b</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="E3:42_1_6_1"/>
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">b</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><b>end;</b></div>
<b>hence </b><a NAME="E2:42_1_6"/>
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">b</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><b>end;</b></div><a NAME="E8:42_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#E2:42_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bvfunc_1.html#E1:42"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E9:42_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#K6" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K6" target="_self">'or'</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#E1:42"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E10:42_1"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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">a</font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">b</font></span>)</span> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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#K6" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">b</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a>, , , <a class="txt" href="bvfunc_1.html#E1:42"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_2"/><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"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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/><a NAME="E2:42_2"/><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  <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="42_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="E1:42_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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="E2:42_2_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<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="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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/>
<a NAME="E3:42_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#K8">TRUE</a>  or <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="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:42_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/>


</div><b>end;</b></div><a NAME="E3:42_2"/><i><font color="Green">E58</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">b</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="42_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="E1:42_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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="E2:42_2_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<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="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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/>
<a NAME="E3:42_2_2"/>
( <font color="Maroon">b</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>  or <font color="Maroon">b</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="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:42_2_2"/>
<font color="Maroon">b</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/>


</div><b>end;</b></div><a NAME="E4:42_2"/>
 <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">b</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#E1:42"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E5:42_2"/><b>then </b><i><font color="Green">E80</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#K6" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#R2">=</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="bvfunc_1.html#K6" target="_self">'or'</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#E1:42"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E6:42_2"/><i><font color="Green">E81</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="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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="42_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_2_3_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"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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="E2:42_2_3_1"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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#D6">VALUAT_1:def 6</a></i>;<br/><a NAME="E3:42_2_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#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E4:42_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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/><b>hence </b><a NAME="E5:42_2_3_1"/>
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="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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>
<b>hence </b><a NAME="E2:42_2_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="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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="E7:42_2"/><i><font color="Green">E83</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  <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="42_2_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_2_4_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> 
 ;<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="E2:42_2_4_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/><b>hence </b><a NAME="E3:42_2_4_1"/>
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><b>end;</b></div>
<b>hence </b><a NAME="E2:42_2_4"/>
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><b>end;</b></div><a NAME="E8:42_2"/><i><font color="Green">E84</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  <font color="Maroon">b</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="42_2_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_2_5_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">b</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="E2:42_2_5_1"/>
<font color="Maroon">b</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/><b>hence </b><a NAME="E3:42_2_5_1"/>
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">b</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><b>end;</b></div>
<b>hence </b><a NAME="E2:42_2_5"/>
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">b</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><b>end;</b></div><a NAME="E9:42_2"/>
 <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#D3" target="_self">Def3</a></i>;<br/><a NAME="E10:42_2"/><b>then </b>
<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="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#R2">=</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="valuat_1.html#K6">'&amp;'</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="E11:42_2"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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">a</font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">b</font></span>)</span> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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#K6" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">b</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="bvfunc_1.html#E1:42"><i><font color="Green">E19</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="hs2(this)" href="javascript:()" title="42_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_3"/><i><font color="Green">E92</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  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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>  &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  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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/><a NAME="E2:42_3"/><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  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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>  ) iff ( ( 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">b</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="42_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:42_3_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  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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#K8">TRUE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_3_1_1"/><i><font color="Green">E95</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="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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="E2:42_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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#D6">VALUAT_1:def 6</a></i>;<br/>
<a NAME="E3:42_3_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<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="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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>thus </b><a NAME="E4:42_3_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="ref" href="xboolean.html#T101">XBOOLEAN:101</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:42_3_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  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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">b</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="42_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_3_1_2"/><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">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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="E2:42_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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="E3:42_3_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<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="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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#D6">VALUAT_1:def 6</a></i>;<br/>
<b>thus </b><a NAME="E4:42_3_1_2"/>
<font color="Maroon">b</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="ref" href="xboolean.html#T101">XBOOLEAN:101</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:42_3_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; ( 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">b</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"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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="42_3_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_3_1_3"/><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  <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">b</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="E2:42_3_1_3"/><i><font color="Green">E101</font></i>: 
<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="E3:42_3_1_3"/>
<font color="Maroon">b</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="E4:42_3_1_3"/><b>then </b>
<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="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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="E5:42_3_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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#D6">VALUAT_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:42_3_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"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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>  ) iff ( ( 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">b</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>  ) ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E3:42_3"/>
( ( 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#K6" target="_self">'or'</a> <font color="Maroon">b</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>  ) iff ( ( 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">b</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="42_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:42_3_2"/><i><font color="Green">E102</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#K6" target="_self">'or'</a> <font color="Maroon">b</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#K7">FALSE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_3_2_1"/><i><font color="Green">E103</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#K6" target="_self">'or'</a> <font color="Maroon">b</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="E2:42_3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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="E3:42_3_2_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<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="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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/>
<a NAME="E4:42_3_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#K8">TRUE</a>  or <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="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>
<b>hence </b><a NAME="E5:42_3_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/>


</div><b>end;</b></div>
<a NAME="E2:42_3_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  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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">b</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="42_3_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_3_2_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">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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="E2:42_3_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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="E3:42_3_2_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<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="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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/>
<a NAME="E4:42_3_2_2"/>
( <font color="Maroon">b</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>  or <font color="Maroon">b</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="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>
<b>hence </b><a NAME="E5:42_3_2_2"/>
<font color="Maroon">b</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/>


</div><b>end;</b></div>
<a NAME="E3:42_3_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>  ) &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">b</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"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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="42_3_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_3_2_3"/><i><font color="Green">E108</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">b</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="E2:42_3_2_3"/><i><font color="Green">E109</font></i>: 
<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="E3:42_3_2_3"/>
<font color="Maroon">b</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="E4:42_3_2_3"/><b>then </b>
<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="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">b</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="E5:42_3_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:42_3_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"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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>  ) iff ( ( 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">b</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>  ) ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E4:42_3"/><b>then </b><i><font color="Green">E110</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> or  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">b</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K16" target="_self">O_el</a> <font color="Maroon">Y</font> ) &amp; (  <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> or  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">b</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>, <a class="txt" href="bvfunc_1.html#E1:42"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E5:42_3"/><b>then </b><i><font color="Green">E111</font></i>: 
<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="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">b</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="E6:42_3"/>
<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#K6" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">b</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>, </i>;<br/><b>hence </b><a NAME="E7:42_3"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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">a</font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">b</font></span>)</span> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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#K6" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">b</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="bvfunc_1.html#D3" target="_self">Def3</a>, <a class="txt" href="bvfunc_1.html#E1:42"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:42"/>
(  <a href="bvfunc_1.html#K18" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">b</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">a</font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">B_INF</a> <font color="Maroon">b</font></span>)</span> &amp;  <a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6" target="_self">'or'</a> <font color="Maroon">b</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#K6" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">B_SUP</a> <font color="Maroon">b</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div>
