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<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> , <font color="Maroon">d</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>assume </b><a NAME="E1:21"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">b</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17">I_el</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17">I_el</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:21"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:21_1"/>
<span class="p1">(<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">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i>, <a class="ref" href="bvfunc_1.html#D14">BVFUNC_1:def 14</a></i>;<br/>
<a NAME="E2:21_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11">BVFUNC_1:def 11</a></i>;<br/>
<a NAME="E3:21_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i>, <a class="ref" href="bvfunc_1.html#D14">BVFUNC_1:def 14</a></i>;<br/>
<a NAME="E4:21_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11">BVFUNC_1:def 11</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#D11">BVFUNC_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="valuat_1.html#D6">VALUAT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<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="margrel1.html#K9">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></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">x</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="margrel1.html#K9">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T23">BINARITH:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></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="margrel1.html#K9">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>

<br/>.= 
<a href="margrel1.html#K8">TRUE</a>  <a href="xboolean.html#K4">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboolean.html#K5">'or'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span>
<b>by </b><i>, , <a class="ref" href="binarith.html#T20">BINARITH:20</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 

;<br/>
<b>hence </b><a NAME="E6:21_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c</font></span>)</span> <a href="bvfunc_1.html#K12">'imp'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K17">I_el</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="bvfunc_1.html#D14">BVFUNC_1:def 14</a></i>;<br/>


</div>
