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

<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">s4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">s5</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">s6</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">s7</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">ns4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">ns5</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">ns6</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">ns7</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">q3</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">nq3</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="E1:4"/><i><font color="Green">E29</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#K8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q3</font></span>)</span>,<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="E2:4"/><i><font color="Green">E30</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#K8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q3</font></span>)</span>,<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="E3:4"/><i><font color="Green">E31</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#K8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q3</font></span>)</span>,<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="E4:4"/><i><font color="Green">E32</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#K8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q3</font></span>)</span>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font> )
 <b>and </b><br/><a NAME="E5:4"/><i><font color="Green">E33</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8">AND3</a> <font color="Maroon">q3</font>,<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="E6:4"/><i><font color="Green">E34</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s5</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8">AND3</a> <font color="Maroon">q3</font>,<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="E7:4"/><i><font color="Green">E35</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s6</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8">AND3</a> <font color="Maroon">q3</font>,<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="E8:4"/><i><font color="Green">E36</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s7</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8">AND3</a> <font color="Maroon">q3</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font> )
 <b>and </b><br/><a NAME="E9:4"/><i><font color="Green">E37</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#K8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq3</font></span>)</span>,<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="E10:4"/><i><font color="Green">E38</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#K8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq3</font></span>)</span>,<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="E11:4"/><i><font color="Green">E39</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#K8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq3</font></span>)</span>,<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="E12:4"/><i><font color="Green">E40</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#K8">AND3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">nq3</font></span>)</span>,<font color="Maroon">nq2</font>,<font color="Maroon">nq1</font> )
 <b>and </b><br/><a NAME="E13:4"/><i><font color="Green">E41</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8">AND3</a> <font color="Maroon">nq3</font>,<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="E14:4"/><i><font color="Green">E42</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns5</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8">AND3</a> <font color="Maroon">nq3</font>,<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="E15:4"/><i><font color="Green">E43</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns6</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8">AND3</a> <font color="Maroon">nq3</font>,<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="E16:4"/><i><font color="Green">E44</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns7</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8">AND3</a> <font color="Maroon">nq3</font>,<font color="Maroon">nq2</font>,<font color="Maroon">nq1</font> )
 <b>and </b><br/><a NAME="E17:4"/><i><font color="Green">E45</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">q1</font></span>)</span>,<font color="Maroon">R</font> )
 <b>and </b><br/><a NAME="E18:4"/><i><font color="Green">E46</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> <span class="p1">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span>,<font color="Maroon">R</font> )
 <b>and </b><br/><a NAME="E19:4"/><i><font color="Green">E47</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K3">OR2</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p3">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p3">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">R</font> )
 ;<br/>

<a NAME="E20:4"/>
(  <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">nq3</font> &amp;  <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">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>, <a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>
<a NAME="E21:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns1</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq3</font> &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</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="E22:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns1</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</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">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span></span>)</span> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</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>, <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<a NAME="E23:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns1</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp; not  <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#T7">GATE_1:7</a></i>;<br/>
<a NAME="E24:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns1</font> iff ( (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font> ) &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font> ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp; not  <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/>
<a NAME="E25:4"/><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">q3</font> &amp;  <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="E26:4"/>
(  <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>, <a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>

<a NAME="E27:4"/>
(  <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#K1">NOT1</a> <font color="Maroon">nq3</font> &amp;  <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#T16">GATE_1:16</a></i>;<br/>
<a NAME="E28:4"/><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> <font color="Maroon">nq3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font> or  not  <a href="gate_1.html#NR1" target="_self">$</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="E29:4"/><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> <font color="Maroon">nq3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T4">GATE_1:4</a></i>;<br/>
<a NAME="E30:4"/><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#K3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span></span>)</span> &amp; not  <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#T6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<a NAME="E31:4"/><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#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" target="_self">$</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#T7">GATE_1:7</a></i>;<br/>
<a NAME="E32:4"/><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> <font color="Maroon">q3</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font> ) &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font> ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</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#T6">GATE_1:6</a></i>;<br/>
<a NAME="E33:4"/><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> <font color="Maroon">q3</font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp; not  <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#T8">GATE_1:8</a></i>;<br/>
<a NAME="E34:4"/><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>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q3</font> &amp;  <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#T4">GATE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E35:4"/>
(  <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">s1</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>

<a NAME="E36:4"/>
(  <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">nq3</font> &amp;  <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">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>, <a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>
<a NAME="E37:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns3</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</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="E38:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns3</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</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">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span></span>)</span> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</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>, <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<a NAME="E39:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns3</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp; not  <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#T7">GATE_1:7</a></i>;<br/>
<a NAME="E40:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns3</font> iff ( (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font> ) &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font> ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp; not  <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/>
<a NAME="E41:4"/><b>then </b>
(  <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">q3</font> &amp;  <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> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T4">GATE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E42:4"/>
(  <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">s2</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>

<a NAME="E43:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns4</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq3</font> &amp;  <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#T16">GATE_1:16</a></i>;<br/>
<a NAME="E44:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns4</font> iff (  <a href="gate_1.html#NR1" target="_self">$</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">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span></span>)</span> &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">nq2</font> &amp; not  <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">q1</font></span>)</span>,<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="E45:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns4</font> iff (  <a href="gate_1.html#NR1" target="_self">$</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">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span></span>)</span> &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">nq2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E46:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns4</font> iff (  <a href="gate_1.html#NR1" target="_self">$</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">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span></span>)</span> &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>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</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="E47:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns4</font> iff ( (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span> or  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span> ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T7">GATE_1:7</a>, <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<a NAME="E48:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns4</font> iff ( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font> ) ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E49:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns4</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp;  <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">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T4">GATE_1:4</a>, <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<a NAME="E50:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns4</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q3</font> &amp;  <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="E51:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns4</font> iff  <a href="gate_1.html#NR1" target="_self">$</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>, <a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>

<a NAME="E52:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns5</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq3</font> &amp;  <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">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>, <a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>
<a NAME="E53:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns5</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq3</font> &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</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="E54:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns5</font> iff (  <a href="gate_1.html#NR1" target="_self">$</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">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span></span>)</span> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</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>, <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<a NAME="E55:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns5</font> iff ( (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span> or  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span> ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp; not  <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#T7">GATE_1:7</a></i>;<br/>
<a NAME="E56:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns5</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp;  <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>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<b>hence </b><a NAME="E57:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns5</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">s4</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>

<a NAME="E58:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns6</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq3</font> &amp;  <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#T16">GATE_1:16</a></i>;<br/>
<a NAME="E59:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns6</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font> or  not  <a href="gate_1.html#NR1" target="_self">$</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="E60:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns6</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T4">GATE_1:4</a></i>;<br/>
<a NAME="E61:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns6</font> iff (  <a href="gate_1.html#NR1" target="_self">$</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">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span></span>)</span> &amp; not  <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">R</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> ) )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<a NAME="E62:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns6</font> iff ( (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span> or  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span> ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T7">GATE_1:7</a></i>;<br/>
<a NAME="E63:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns6</font> iff ( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font> ) ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">R</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E64:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns6</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</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">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<a NAME="E65:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns6</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp;  <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#T4">GATE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E66:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns6</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">s5</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>

<a NAME="E67:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns7</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq3</font> &amp;  <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">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>, <a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>
<a NAME="E68:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns7</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">nq3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</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="E69:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns7</font> iff (  <a href="gate_1.html#NR1" target="_self">$</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">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span></span>)</span> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</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>, <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<a NAME="E70:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns7</font> iff ( (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span> or  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span> ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp; not  <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#T7">GATE_1:7</a></i>;<br/>
<a NAME="E71:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns7</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp;  <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> ) )
 
<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="E72:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">ns7</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">s6</font>,<font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>

<a NAME="E73:4"/>
(  <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">nq3</font> &amp;  <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#T16">GATE_1:16</a></i>;<br/>
<a NAME="E74:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p2">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span></span>)</span> ) or  not  <a href="gate_1.html#NR1" target="_self">$</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>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E75:4"/><b>then </b>
( ( (  <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">q2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q1</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span> ) 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#T7">GATE_1:7</a>, <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<a NAME="E76:4"/><b>then </b>
( ( (  <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">q2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">q3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">q1</font></span>)</span> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</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#T6">GATE_1:6</a></i>;<br/>
<a NAME="E77:4"/><b>then </b>
( ( (  <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">q2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> ) or (  <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">q2</font> &amp; not  <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">q3</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#T6">GATE_1:6</a>, <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<a NAME="E78:4"/><b>then </b>
( ( (  <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">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">q3</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="E79:4"/><b>then </b>
( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s7</font> or  <a href="gate_1.html#NR1" target="_self">$</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#T16">GATE_1:16</a></i>;<br/>
<b>hence </b><a NAME="E80:4"/>
(  <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> <font color="Maroon">s7</font>,<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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