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<b>let </b><font color="Maroon">s0</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">s1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">s2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">s3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">ns0</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">ns1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">ns2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">ns3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">q1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">q2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">nq1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">nq2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">R</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:2"/><i><font color="Green">E14</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s0</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:2"/><i><font color="Green">E15</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s1</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q2</font></span>)</span>,<font color="Maroon">q1</font> )
 <b>and </b><br/><a NAME="E3:2"/><i><font color="Green">E16</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:2"/><i><font color="Green">E17</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font> )
 <b>and </b><br/><a NAME="E5:2"/><i><font color="Green">E18</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns0</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq1</font></span>)</span> )
 <b>and </b><br/><a NAME="E6:2"/><i><font color="Green">E19</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns1</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq2</font></span>)</span>,<font color="Maroon">nq1</font> )
 <b>and </b><br/><a NAME="E7:2"/><i><font color="Green">E20</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">nq2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq1</font></span>)</span> )
 <b>and </b><br/><a NAME="E8:2"/><i><font color="Green">E21</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">nq2</font>,<font color="Maroon">nq1</font> )
 <b>and </b><br/><a NAME="E9:2"/><i><font color="Green">E22</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq1</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q2</font></span>)</span>,<font color="Maroon">R</font> )
 <b>and </b><br/><a NAME="E10:2"/><i><font color="Green">E23</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<font color="Maroon">R</font> )
 ;<br/>

<a NAME="E11:2"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns1</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq1</font> ) )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E12:2"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns1</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) ) )
 
<b>by </b><i>, , <a class="ref" href="gate_1.html#T4">GATE_1:4</a>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E13:2"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns1</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T4">GATE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E14:2"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns1</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">s0</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>

<a NAME="E15:2"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns3</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) )
 
<b>by </b><i>, , , <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<b>hence </b><a NAME="E16:2"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">s1</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>

<a NAME="E17:2"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns2</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq1</font> ) )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E18:2"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns2</font> iff ( (  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q2</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) )
 
<b>by </b><i>, , <a class="ref" href="gate_1.html#T4">GATE_1:4</a>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E19:2"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns2</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T4">GATE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E20:2"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">s3</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>

<a NAME="E21:2"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns0</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq1</font> ) )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E22:2"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns0</font> iff ( (  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q2</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) ) )
 
<b>by </b><i>, , <a class="ref" href="gate_1.html#T4">GATE_1:4</a>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E23:2"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns0</font> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T4">GATE_1:4</a></i>;<br/>
<a NAME="E24:2"/><b>then </b>
( (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">s2</font>,<font color="Maroon">R</font> or  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">R</font> ) iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns0</font> )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#T4">GATE_1:4</a>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<b>hence </b><a NAME="E25:2"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns0</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">s2</font>,<font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">R</font></span>)</span> )
 <b>by </b><i><a class="ref" href="gate_1.html#T7">GATE_1:7</a></i>;<br/>


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