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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>



<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:48_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:48_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>;<br/>
<a NAME="E2:48_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:48_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E2:48_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="margrel1.html#D17">MARGREL1:def 17</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:48_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:48_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E1:48_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E2:48_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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="E3:48_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E2:48_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="margrel1.html#D17">MARGREL1:def 17</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:48_1"/><i><font color="Green">E30</font></i>: 
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><i><font color="Green">E31</font></i>: <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="bvfunc_1.html#E3:48_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="bvfunc_1.html#E2:48_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>
;<br/><a NAME="E6:48_1"/><i><font color="Green">E32</font></i>: 
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_4_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:48_1_4_1"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E3:48_1_4_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E2:48_1"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E4:48_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E2:48_1_4_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/><b>hence </b><a NAME="E5:48_1_4_1"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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="E2:48_1_4"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E7:48_1"/><i><font color="Green">E33</font></i>: 
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_5_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:48_1_5_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E2:48_1"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:48_1_5_1"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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="E2:48_1_5"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E8:48_1"/><i><font color="Green">E34</font></i>: 
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_6_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:48_1_6_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E3:48_1"><i><font color="Green">E29</font></i></a></i>;<br/><b>hence </b><a NAME="E3:48_1_6_1"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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="E2:48_1_6"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E9:48_1"/>
 <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E7:48_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E10:48_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E8:48_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E11:48_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#T10" target="_self">Th10</a></i>;<br/><b>hence </b><a NAME="E12:48_1"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E4:48_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:48_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bvfunc_1.html#E6:48_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a>, <a class="ref" href="bvfunc_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:48_2"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:48_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E2:48_2_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:48_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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="E4:48_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:48_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:48_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:48_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E2:48_2_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:48_2_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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="E4:48_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:48_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:48_2"/><i><font color="Green">E31</font></i>: 
 <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E5:48_2"/>
 <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E3:48_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E6:48_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:48_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E7:48_2"/><i><font color="Green">E33</font></i>: 
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:48_2_3_1"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>;<br/><a NAME="E3:48_2_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:48_2"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E4:48_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:48_2_3_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><b>hence </b><a NAME="E5:48_2_3_1"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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="E2:48_2_3"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E8:48_2"/><i><font color="Green">E34</font></i>: 
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2_4_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:48_2_4_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E2:48_2"><i><font color="Green">E29</font></i></a></i>;<br/><b>hence </b><a NAME="E3:48_2_4_1"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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="E2:48_2_4"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E9:48_2"/><i><font color="Green">E35</font></i>: 
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2_5_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:48_2_5_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E3:48_2"><i><font color="Green">E30</font></i></a></i>;<br/><b>hence </b><a NAME="E3:48_2_5_1"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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="E2:48_2_5"/>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E10:48_2"/>
 <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E8:48_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E11:48_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></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">O_el</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E9:48_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a></i>;<br/><a NAME="E12:48_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E13:48_2"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E4:48_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bvfunc_1.html#E6:48_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="bvfunc_1.html#E7:48_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a>, <a class="ref" href="bvfunc_1.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3"/><i><font color="Green">E28</font></i>: 
( not <!--HUMANRECFAILED-->for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; not <!--HUMANRECFAILED-->for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 ;<br/><a NAME="E2:48_3"/><i><font color="Green">E29</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) iff ( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_3_1"/><i><font color="Green">E30</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_3_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:48_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>;<br/>
<a NAME="E3:48_3_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_3_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:48_3_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E3:48_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="margrel1.html#D17">MARGREL1:def 17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:48_3_1"/><i><font color="Green">E31</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_3_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:48_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E1:48_3_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E3:48_3_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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="E4:48_3_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E3:48_3_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="margrel1.html#D17">MARGREL1:def 17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:48_3_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_3_1_3"/><i><font color="Green">E32</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:48_3_1_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E1:48_3_1_3"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E3:48_3_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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#E1:48_3_1_3"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E4:48_3_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E2:48_3_1_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="margrel1.html#D17">MARGREL1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E5:48_3_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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="E4:48_3_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) iff ( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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#E1:48_3_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:48_3_1"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:48_3"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) iff ( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:48_3_2"/><i><font color="Green">E30</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_3_2_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:48_3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_3_2_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E3:48_3_2_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E4:48_3_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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="E5:48_3_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E3:48_3_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:48_3_2"/><i><font color="Green">E31</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_3_2_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:48_3_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_3_2_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E3:48_3_2_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E4:48_3_2_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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="E5:48_3_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E3:48_3_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:48_3_2"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_3_2_3"/><i><font color="Green">E32</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:48_3_2_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_3_2_3"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E3:48_3_2_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_3_2_3"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E4:48_3_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="bvfunc_1.html#E2:48_3_2_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="binarith.html#T7">BINARITH:7</a></i>;<br/>
<b>hence </b><a NAME="E5:48_3_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:48_3_2"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) iff ( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  ) ) )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_3_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:48_3_2"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:48_3"/><b>then </b><i><font color="Green">E30</font></i>: 
( (  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; (  <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:48_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E5:48_3"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K18" target="_self">O_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E6:48_3"/>
<span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K19" target="_self">I_el</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E4:48_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="bvfunc_1.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E7:48_3"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="bvfunc_1.html#E5:48_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="bvfunc_1.html#D16" target="_self">Def16</a>, <a class="ref" href="bvfunc_1.html#D17" target="_self">Def17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48"/>
(  <a href="bvfunc_1.html#K20" target="_self">B_INF</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K20" target="_self">B_INF</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K8" target="_self">'or'</a> <span class="p1">(<span class="default"><a href="bvfunc_1.html#K21" target="_self">B_SUP</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="bvfunc_1.html#E1:48"><i><font color="Green">E26</font></i></a>, <a class="txt" href="bvfunc_1.html#E2:48"><i><font color="Green">E27</font></i></a></i>;<br/>


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