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<b>let </b><font color="Maroon" title="c1">s0</font>, <font color="Maroon" title="c2">s1</font>, <font color="Maroon" title="c3">s2</font>, <font color="Maroon" title="c4">s3</font>, <font color="Maroon" title="c5">ns0</font>, <font color="Maroon" title="c6">ns1</font>, <font color="Maroon" title="c7">ns2</font>, <font color="Maroon" title="c8">ns3</font>, <font color="Maroon" title="c9">q1</font>, <font color="Maroon" title="c10">q2</font>, <font color="Maroon" title="c11">nq1</font>, <font color="Maroon" title="c12">nq2</font>, <font color="Maroon" title="c13">R</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 <!--HUMANRECFAILED-->( ( not <font color="Maroon" title="c1">s0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c1">s0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c2">s1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font></span>)</span>,<font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font></span>)</span>,<font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c2">s1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c3">s2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">q2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">q2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c3">s2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c4">s3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">q2</font>,<font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">q2</font>,<font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c4">s3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c6">ns1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font></span>)</span>,<font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font></span>)</span>,<font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c6">ns1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">nq2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">nq2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c8">ns3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">nq2</font>,<font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">nq2</font>,<font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c8">ns3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font></span>)</span>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font></span>)</span>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c12">nq2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">q1</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">q1</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c12">nq2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <!--HUMANRECFAILED-->( ( not <font color="Maroon" title="c6">ns1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">s0</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">s0</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c6">ns1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c8">ns3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c8">ns3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c5">ns0</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> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c13">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c13">R</font></span>)</span> 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> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c13">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c13">R</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c5">ns0</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="E1:2"/><i><font color="Green" title="E1">A1</font></i>: 
( not <font color="Maroon" title="c1">s0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E2:2"/><i><font color="Green" title="E2">A2</font></i>: 
( not <font color="Maroon" title="c2">s1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font></span>)</span>,<font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E3:2"/><i><font color="Green" title="E3">A3</font></i>: 
( not <font color="Maroon" title="c3">s2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">q2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E4:2"/><i><font color="Green" title="E4">A4</font></i>: 
( not <font color="Maroon" title="c4">s3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">q2</font>,<font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E5:2"/><i><font color="Green" title="E5">A5</font></i>: 
( not <font color="Maroon" title="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E6:2"/><i><font color="Green" title="E6">A6</font></i>: 
( not <font color="Maroon" title="c6">ns1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font></span>)</span>,<font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E7:2"/><i><font color="Green" title="E7">A7</font></i>: 
( not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">nq2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E8:2"/><i><font color="Green" title="E8">A8</font></i>: 
( not <font color="Maroon" title="c8">ns3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">nq2</font>,<font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E9:2"/><i><font color="Green" title="E9">A9</font></i>: 
( not <font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font></span>)</span>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E10:2"/><i><font color="Green" title="E10">A10</font></i>: 
( not <font color="Maroon" title="c12">nq2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c9">q1</font>,<font color="Maroon" title="c13">R</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="c6">ns1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">s0</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">s0</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c6">ns1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c8">ns3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c8">ns3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c5">ns0</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> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c13">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c13">R</font></span>)</span> 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> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c13">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c13">R</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )</span><br/>

<a NAME="E11:2"/>
( not <font color="Maroon" title="c6">ns1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 
<b>by </b><i><a class="txt" href="#E6:2"><i><font color="Green" title="E6">A6</font></i></a></i>;<br/>
<a NAME="E12:2"/><b>then </b>
( not <font color="Maroon" title="c6">ns1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; ( <font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) )
 
<b>by </b><i><a class="txt" href="#E9:2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="#E10:2"><i><font color="Green" title="E10">A10</font></i></a></i>;<br/>
<a NAME="E13:2"/><b>then </b>
( not <font color="Maroon" title="c6">ns1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 
;<br/>
<b>hence </b><a NAME="E14:2"/>
( not <font color="Maroon" title="c6">ns1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">s0</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E1:2"><i><font color="Green" title="E1">A1</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="c8">ns3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c8">ns3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c5">ns0</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> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c13">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c13">R</font></span>)</span> 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> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c13">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c13">R</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )</span><br/>

<a NAME="E15:2"/>
( not <font color="Maroon" title="c8">ns3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 
<b>by </b><i><a class="txt" href="#E8:2"><i><font color="Green" title="E8">A8</font></i></a>, <a class="txt" href="#E9:2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="#E10:2"><i><font color="Green" title="E10">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:2"/>
( not <font color="Maroon" title="c8">ns3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">s1</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E2:2"><i><font color="Green" title="E2">A2</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="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c5">ns0</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> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c13">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c13">R</font></span>)</span> 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> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c13">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c13">R</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )</span><br/>

<a NAME="E17:2"/>
( not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( not <font color="Maroon" title="c12">nq2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 
<b>by </b><i><a class="txt" href="#E7:2"><i><font color="Green" title="E7">A7</font></i></a></i>;<br/>
<a NAME="E18:2"/><b>then </b>
( not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( (  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 
<b>by </b><i><a class="txt" href="#E9:2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="#E10:2"><i><font color="Green" title="E10">A10</font></i></a></i>;<br/>
<a NAME="E19:2"/><b>then </b>
( not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( not <font color="Maroon" title="c10">q2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 
;<br/>
<b>hence </b><a NAME="E20:2"/>
( not <font color="Maroon" title="c7">ns2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c4">s3</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E4:2"><i><font color="Green" title="E4">A4</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="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c13">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c13">R</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/>

<a NAME="E21:2"/>
( not <font color="Maroon" title="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 
<b>by </b><i><a class="txt" href="#E5:2"><i><font color="Green" title="E5">A5</font></i></a></i>;<br/>
<a NAME="E22:2"/><b>then </b>
( not <font color="Maroon" title="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( (  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( <font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) )
 
<b>by </b><i><a class="txt" href="#E9:2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="#E10:2"><i><font color="Green" title="E10">A10</font></i></a></i>;<br/>
<a NAME="E23:2"/><b>then </b>
( ( ( not <font color="Maroon" title="c10">q2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) iff  not <font color="Maroon" title="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
;<br/>
<a NAME="E24:2"/><b>then </b>
( (  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c13">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) iff  not <font color="Maroon" title="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="#E3:2"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E25:2"/>
( not <font color="Maroon" title="c5">ns0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K3" title="GATE_1:func.3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">s2</font>,<font color="Maroon" title="c13">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c13">R</font></span>)</span> 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/>


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