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<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>assume </b><a NAME="E1:9"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K17">I_el</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">b</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="bvfunc_1.html#K17">I_el</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:9"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</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="9_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="E1:9_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_1.html#K1">.</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="E2:9_1"/><b>then </b><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><a href="xboolean.html#K3">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</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_1.html#K1">.</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="E3:9_1"/><i><font color="Green">E17</font></i>: 
(  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>
<a NAME="E4:9_1"/>
<span class="p1">(<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_1.html#K1">.</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="E5:9_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><a href="xboolean.html#K3">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</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">c</font> <a href="funct_1.html#K1">.</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="E6:9_1"/><i><font color="Green">E20</font></i>: 
(  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  or  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>
<a NAME="E7:9_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xboolean.html#K3">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</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">c</font> <a href="funct_1.html#K1">.</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_1"/>
( (  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) or (  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) or ( <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) or ( <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 </span><b>by </b><i>, , , , <a class="ref" href="binarith.html#T7">BINARITH:7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1"><b>case </b></a><a NAME="E1:9_1_1_1_1"/>
(  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</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="binarith.html#T19">BINARITH:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_2"><b>case </b></a><a NAME="E1:9_1_1_1_2"/>
(  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_3"><b>case </b></a><a NAME="E1:9_1_1_1_3"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="xboolean.html#K3">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div class="add"><a NAME="E2:9_1_1_1_3"/><b>then </b>
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_4"><b>case </b></a><a NAME="E1:9_1_1_1_4"/>
( <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</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="binarith.html#T19">BINARITH:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:9_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</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="E3:9"/>
<font color="Maroon">a</font> <a href="bvfunc_1.html#K12">'imp'</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</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/>


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