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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green" title="E34">A1</font></i>: 
( 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> )
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green" title="E35">A2</font></i>: 
( 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> )
 <b>and </b><br/><a NAME="E3:62"/><i><font color="Green" title="E36">A3</font></i>: 
( 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> )
 <b>and </b><br/><a NAME="E4:62"/><i><font color="Green" title="E37">A4</font></i>: 
( 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> )
 <b>and </b><br/><a NAME="E5:62"/><i><font color="Green" title="E38">A5</font></i>: 
( 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> )
 <b>and </b><br/><a NAME="E6:62"/><i><font color="Green" title="E39">A6</font></i>: 
( 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> )
 <b>and </b><br/><a NAME="E7:62"/><i><font color="Green" title="E40">A7</font></i>: 
( 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> )
 <b>and </b><br/><a NAME="E8:62"/><i><font color="Green" title="E41">A8</font></i>: 
( 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> )
 <b>and </b><br/><a NAME="E9:62"/><i><font color="Green" title="E42">A9</font></i>: 
( 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> )
 <b>and </b><br/><a NAME="E10:62"/><i><font color="Green" title="E43">A10</font></i>: 
( 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> )
 <b>and </b><br/><a NAME="E11:62"/><i><font color="Green" title="E44">A11</font></i>: 
( 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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E12:62"/>
( 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> )
 <b>by </b><i><a class="txt" href="#E11:62"><i><font color="Green" title="E44">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E13:62"/><i><font color="Green" title="E45">B12</font></i>: 
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:()"><i><font color="Red">::  thesis: </font></i></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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1"/>
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:()"><i><font color="Red">::  thesis: </font></i></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="E2:62_1"/><b>then </b>
( not <font color="Maroon" title="c1">c1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; (  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> ) &amp; not <font color="Maroon" title="c15">n2</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> &amp; not <font color="Maroon" title="c17">n4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E5:62"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="gate_1.html#D18" target="_self" title="GATE_1:def.18">Def18</a>, <a class="txt" href="#E9:62"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/><a NAME="E3:62_1"/><b>then </b>
( not <font color="Maroon" title="c10">c2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; (  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> ) &amp; not <font color="Maroon" title="c16">n3</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> )
 <b>by </b><i><a class="txt" href="#E6:62"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="#E1:62"><i><font color="Green" title="E34">A1</font></i></a></i>;<br/><a NAME="E4:62_1"/><b>then </b>
( not <font color="Maroon" title="c11">c3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; (  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> ) &amp; not <font color="Maroon" title="c17">n4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E7:62"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="#E2:62"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/><a NAME="E5:62_1"/><b>then </b>
( not <font color="Maroon" title="c12">c4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; (  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> ) )
 <b>by </b><i><a class="txt" href="#E8:62"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="#E3:62"><i><font color="Green" title="E36">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E6:62_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:62"/>
not <font color="Maroon" title="c13">c5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="#E13:62"><i><font color="Green" title="E45">B12</font></i></a>, <a class="txt" href="#E10:62"><i><font color="Green" title="E43">A10</font></i></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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