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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> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">Y</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><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">E22</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"><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#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">E23</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"><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#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"><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="valuat_1.html#K6">'&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">a</font></span>)</span> <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>
<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="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <span class="p4">(<span class="default"><a href="valuat_1.html#K5">'not'</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="bvfunc_1.html#K6">'or'</a> <font color="Maroon">c</font></span>)</span></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="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">a</font> <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="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="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"><font color="Maroon">a</font> <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"><font color="Maroon">a</font> <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"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:22_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <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> <a href="hidden.html#R1">=</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 NAME="E3:22_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<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="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> <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">BVFUNC_1:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">az</font> = <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> as  <a href="xboolean.html#V1">boolean</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E5:22_2"/>
( <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/><a NAME="E6:22_2"/><b>then </b>
( <font color="Maroon">b</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>  &amp; <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/>;<br/><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"><font color="Maroon">a</font> <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"><a href="xboolean.html#K3">'not'</a> <font color="Maroon">az</font></span>)</span> <a href="xboolean.html#K4">'&amp;'</a> <font color="Maroon">az</font>
<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="E8:22_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


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