<?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> , <font color="Maroon">c</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="bvfunc_1.html#D15">BVFUNC_1:def 15</a><br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">b</font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>

<i><font color="Green">E17</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">b</font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_4.html#T8">BVFUNC_4:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_4.html#T8">BVFUNC_4:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xboolean.html#K4">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7">BVFUNC_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xboolean.html#K4">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7">BVFUNC_1:def 7</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">b</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><i><font color="Green">E19</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">b</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="bvfunc_4.html#T8">BVFUNC_4:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="bvfunc_1.html#T16">BVFUNC_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D7">BVFUNC_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>
;<br/><a NAME="E3:22_2"/><i><font color="Green">E20</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <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/><a NAME="E4:22_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or <font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2_2_1"/>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or <span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 </span><b>by </b><i>, , , <a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_1_1"><b>case </b></a><a NAME="E1:22_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xboolean.html#K4">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="valuat_1.html#D5">VALUAT_1:def 5</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="ref" href="xboolean.html#T138">XBOOLEAN:138</a></i>
;<br/><b>hence </b><a NAME="E3:22_2_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">b</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_1_2"><b>case </b></a><a NAME="E1:22_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xboolean.html#K4">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xboolean.html#K4">'&amp;'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, , <a class="ref" href="margrel1.html#T43">MARGREL1:43</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 

;<br/><b>hence </b><a NAME="E3:22_2_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">b</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:22_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">b</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</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="E4:22"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">b</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div>
