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<span class="kw">let </span><font color="Maroon" title="c1">c1</font>, <font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">x3</font>, <font color="Maroon" title="c5">x4</font>, <font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</font>, <font color="Maroon" title="c8">y3</font>, <font color="Maroon" title="c9">y4</font>, <font color="Maroon" title="c10">c2</font>, <font color="Maroon" title="c11">c3</font>, <font color="Maroon" title="c12">c4</font>, <font color="Maroon" title="c13">c5</font>, <font color="Maroon" title="c14">n1</font>, <font color="Maroon" title="c15">n2</font>, <font color="Maroon" title="c16">n3</font>, <font color="Maroon" title="c17">n4</font>, <font color="Maroon" title="c18">n</font>, <font color="Maroon" title="c19">c5b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c1">c1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c10">c2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c10">c2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c11">c3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c4">x3</font>,<font color="Maroon" title="c8">y3</font>,<font color="Maroon" title="c11">c3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c12">c4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c5">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c12">c4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c13">c5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c14">n1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c15">n2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c7">y2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c16">n3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c4">x3</font>,<font color="Maroon" title="c8">y3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c17">n4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c5">x4</font>,<font color="Maroon" title="c9">y4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c18">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K18" title="GATE_1:func.18">AND5</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c14">n1</font>,<font color="Maroon" title="c15">n2</font>,<font color="Maroon" title="c16">n3</font>,<font color="Maroon" title="c17">n4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not <font color="Maroon" title="c19">c5b</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c13">c5</font>,<font color="Maroon" title="c18">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c13">c5</font>,<font color="Maroon" title="c18">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c19">c5b</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) implies ( not <font color="Maroon" title="c13">c5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not <font color="Maroon" title="c19">c5b</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
( ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c1">c1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c10">c2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c10">c2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c11">c3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) &amp; ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c4">x3</font>,<font color="Maroon" title="c8">y3</font>,<font color="Maroon" title="c11">c3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c12">c4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  ) )
 <span class="kw">and </span><br/><a NAME="E2:64"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c5">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c12">c4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c13">c5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E3:64"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
( not <font color="Maroon" title="c14">n1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E4:64"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
( not <font color="Maroon" title="c15">n2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c7">y2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E5:64"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
( not <font color="Maroon" title="c16">n3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c4">x3</font>,<font color="Maroon" title="c8">y3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E6:64"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
( not <font color="Maroon" title="c17">n4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c5">x4</font>,<font color="Maroon" title="c9">y4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E7:64"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
( not <font color="Maroon" title="c18">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K18" title="GATE_1:func.18">AND5</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c14">n1</font>,<font color="Maroon" title="c15">n2</font>,<font color="Maroon" title="c16">n3</font>,<font color="Maroon" title="c17">n4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E8:64"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
( not <font color="Maroon" title="c19">c5b</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c13">c5</font>,<font color="Maroon" title="c18">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E9:64"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
( not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <font color="Maroon" title="c13">c5</font>,<font color="Maroon" title="c18">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c19">c5b</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c13">c5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  not <font color="Maroon" title="c19">c5b</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>

<div><span class="lab"><font color="Green" title="E43">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:64_1"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
not <font color="Maroon" title="c18">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c5">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c12">c4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/><a NAME="E2:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A12</font></span>: 
not <font color="Maroon" title="c1">c1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="gate_1.html#D18" target="_self" title="GATE_1:def.18">Def18</a></span>;<br/><a NAME="E3:64_1"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
(  not <font color="Maroon" title="c4">x3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c8">y3</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="gate_1.html#D18" target="_self" title="GATE_1:def.18">Def18</a></span>;<br/><a NAME="E4:64_1"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
(  not <font color="Maroon" title="c3">x2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c7">y2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="gate_1.html#D18" target="_self" title="GATE_1:def.18">Def18</a></span>;<br/><a NAME="E5:64_1"/><span class="lab"><font color="Green" title="E48">A15</font></span>: 
(  not <font color="Maroon" title="c5">x4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c9">y4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="gate_1.html#D18" target="_self" title="GATE_1:def.18">Def18</a></span>;<br/><a NAME="E6:64_1"/>
(  not <font color="Maroon" title="c2">x1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c6">y1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="gate_1.html#D18" target="_self" title="GATE_1:def.18">Def18</a></span>;<br/><span class="kw">hence </span><a NAME="E7:64_1"/>
not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c5">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c12">c4</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="txt" href="#E4:64_1"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E3:64_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E48">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E11:64"/>
( not <font color="Maroon" title="c13">c5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c19">c5b</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64"><span class="lab"><font color="Green" title="E42">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c19">c5b</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  not <font color="Maroon" title="c13">c5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E12:64"/>
not <font color="Maroon" title="c19">c5b</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c13">c5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<span class="kw">hence </span><a NAME="E13:64"/>
not <font color="Maroon" title="c13">c5</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E8:64"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E10:64"><span class="lab"><font color="Green" title="E43">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
