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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">b1</font>, <font color="Olive" title="b4">b2</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Olive" title="b3">b1</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">a2</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Olive" title="b4">b2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="bvfunc_1.html#K6" title="BVFUNC_1:func.6">'or'</a> <font color="Olive" title="b2">a2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <font color="Olive" title="b4">b2</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Olive" title="b1">a1</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">b2</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Olive" title="b2">a2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="bvfunc_1.html#K6" title="BVFUNC_1:func.6">'or'</a> <font color="Olive" title="b4">b2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <font color="Olive" title="b2">a2</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c4">b1</font>, <font color="Maroon" title="c5">b2</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">Y</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a2</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="bvfunc_1.html#K6" title="BVFUNC_1:func.6">'or'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c2">a1</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="bvfunc_1.html#K6" title="BVFUNC_1:func.6">'or'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span></span><br/>



<a NAME="E1:10"/><i><font color="Green" title="E4">A1</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a2</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="bvfunc_1.html#K6" title="BVFUNC_1:func.6">'or'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c2">a1</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="bvfunc_1.html#K6" title="BVFUNC_1:func.6">'or'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="bvfunc10.html#E9"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<a NAME="E2:10"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c2">a1</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="bvfunc_1.html#K6" title="BVFUNC_1:func.6">'or'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span> <a href="bvfunc_1.html#R1" title="BVFUNC_1:pred.1">'&lt;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a2</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="bvfunc_1.html#K6" title="BVFUNC_1:func.6">'or'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="bvfunc10.html#E9"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:10"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a2</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="bvfunc_1.html#K6" title="BVFUNC_1:func.6">'or'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c2">a1</font></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="bvfunc_1.html#K12" title="BVFUNC_1:func.12">'imp'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="bvfunc_1.html#K6" title="BVFUNC_1:func.6">'or'</a> <font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> <font color="Maroon" title="c3">a2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc10.html#E1:10"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="bvfunc_1.html#T18" title="BVFUNC_1:th.18">BVFUNC_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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