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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></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">E3</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>9</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E4</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>10</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green">E5</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>11</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E4:4"/><i><font color="Green">E6</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>12</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E5:4"/><i><font color="Green">E7</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>13</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E6:4"/><i><font color="Green">E8</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>5</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E7:4"/><i><font color="Green">E9</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>6</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E8:4"/><i><font color="Green">E10</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>7</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E9:4"/><i><font color="Green">E11</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>8</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E10:4"/><i><font color="Green">E12</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>12</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="gate_5.html#E4:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="gate_5.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E11:4"/><i><font color="Green">E13</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>13</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="gate_5.html#E5:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="gate_5.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E12:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>5</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K1" target="_self">MULT210</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="gate_5.html#E1:4"><i><font color="Green">E3</font></i></a>, <a class="txt" href="gate_5.html#E6:4"><i><font color="Green">E8</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>6</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K2" target="_self">MULT211</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="gate_5.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="txt" href="gate_5.html#E7:4"><i><font color="Green">E9</font></i></a></i>;<br/>

<b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="gate_5.html#K3" target="_self">MULT212</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
 <a href="gate_5.html#K3" target="_self">MULT212</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
 <a href="gate_1.html#K35">ADD2</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> 

<br/>.= 
 <a href="gate_1.html#K10">XOR3</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K11">MAJ3</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="gate_1.html#D37">GATE_1:def 37</a></i>
;<br/>
<a NAME="E15:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K3" target="_self">MULT212</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K4">XOR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K11">MAJ3</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_5.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E16:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a>  ) or ( not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a>  ) ) iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K3" target="_self">MULT212</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E17:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>7</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K3" target="_self">MULT212</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="gate_5.html#E3:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="gate_5.html#E8:4"><i><font color="Green">E10</font></i></a>, <a class="txt" href="gate_5.html#E10:4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>

<b>set </b><font color="Maroon">c<sub>18</sub></font> =  <a href="gate_5.html#K4" target="_self">MULT213</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
 <a href="gate_5.html#K4" target="_self">MULT213</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
 <a href="gate_1.html#K36">CARR2</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> 

<br/>.= 
 <a href="gate_1.html#K11">MAJ3</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K11">MAJ3</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="gate_1.html#D38">GATE_1:def 38</a></i>
;<br/>
<a NAME="E19:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K4" target="_self">MULT213</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K11">MAJ3</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_5.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E20:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K4" target="_self">MULT213</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> iff (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="xboole_0.html#K1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<b>hence </b><a NAME="E21:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>8</sub></font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K4" target="_self">MULT213</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="gate_5.html#E3:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="gate_5.html#E9:4"><i><font color="Green">E11</font></i></a>, <a class="txt" href="gate_5.html#E11:4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>


</div>
