<?xml version="1.0"?>
<div class="add">

<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> 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-->( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">s0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">s0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">s1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font></span>)</span>,<font color="Maroon" title="c9">q1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">s1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">q2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c10">q2</font>,<font color="Maroon" title="c9">q1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">ns1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font></span>)</span>,<font color="Maroon" title="c11">nq1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">ns1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">nq2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font></span>)</span> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">ns3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c12">nq2</font>,<font color="Maroon" title="c11">nq1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">ns3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">nq1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">nq1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">nq2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">q1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">q1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">nq2</font> ) &amp; not <!--HUMANRECFAILED-->( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">ns1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">s0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">s0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">ns1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">ns3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">s1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">s1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">ns3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> ) ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1"/><i><font color="Green" title="E1">A1</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">s0</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> )
 <b>and </b><br/><a NAME="E2:1"/><i><font color="Green" title="E2">A2</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">s1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> )
 <b>and </b><br/><a NAME="E3:1"/><i><font color="Green" title="E3">A3</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> )
 <b>and </b><br/><a NAME="E4:1"/><i><font color="Green" title="E4">A4</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> )
 <b>and </b><br/><a NAME="E5:1"/><i><font color="Green" title="E5">A5</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> )
 <b>and </b><br/><a NAME="E6:1"/><i><font color="Green" title="E6">A6</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">ns1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> )
 <b>and </b><br/><a NAME="E7:1"/><i><font color="Green" title="E7">A7</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> )
 <b>and </b><br/><a NAME="E8:1"/><i><font color="Green" title="E8">A8</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">ns3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <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> )
 <b>and </b><br/><a NAME="E9:1"/><i><font color="Green" title="E9">A9</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">nq1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font> )
 <b>and </b><br/><a NAME="E10:1"/><i><font color="Green" title="E10">A10</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">nq2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">q1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">ns1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">s0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">s0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">ns1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">ns3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">s1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">s1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">ns3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> ) )</span><br/>

<a NAME="E11:1"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">ns1</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">nq1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_3.html#E6:1"><i><font color="Green" title="E6">A6</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E12:1"/><b>then </b>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">ns1</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c10">q2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_3.html#E9:1"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="gate_3.html#E10:1"><i><font color="Green" title="E10">A10</font></i></a>, <a class="ref" href="gate_1.html#T4" title="GATE_1:th.4">GATE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E13:1"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">ns1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">s0</font> )
 <b>by </b><i><a class="txt" href="gate_3.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">ns3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">s1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">s1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">ns3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> ) )</span><br/>

<b>thus </b><a NAME="E14:1"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">ns3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">s1</font> )
 <b>by </b><i><a class="txt" href="gate_3.html#E2:1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="gate_3.html#E8:1"><i><font color="Green" title="E8">A8</font></i></a>, <a class="txt" href="gate_3.html#E9:1"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="gate_3.html#E10:1"><i><font color="Green" title="E10">A10</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> ) )</span><br/>

<a NAME="E15:1"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">nq2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_3.html#E7:1"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E16:1"/><b>then </b>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">q2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">q1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_3.html#E9:1"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="gate_3.html#E10:1"><i><font color="Green" title="E10">A10</font></i></a>, <a class="ref" href="gate_1.html#T4" title="GATE_1:th.4">GATE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E17:1"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">ns2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">s3</font> )
 <b>by </b><i><a class="txt" href="gate_3.html#E4:1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> )</span><br/>

<a NAME="E18:1"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c12">nq2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c11">nq1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_3.html#E5:1"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<a NAME="E19:1"/><b>then </b>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">q2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c9">q1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_3.html#E9:1"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="gate_3.html#E10:1"><i><font color="Green" title="E10">A10</font></i></a>, <a class="ref" href="gate_1.html#T4" title="GATE_1:th.4">GATE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E20:1"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">ns0</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">s2</font> )
 <b>by </b><i><a class="txt" href="gate_3.html#E3:1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
