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<b>let </b><font color="Maroon">x0</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y0</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z0</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">q00</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">q01</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c01</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">q11</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c11</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">E21</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q00</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">x0</font>,<font color="Maroon">y0</font> )
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E22</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q01</font> iff  <a href="gate_1.html#NR1" target="_self">$</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">x1</font>,<font color="Maroon">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">x0</font>,<font color="Maroon">y1</font></span>)</span>,<a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green">E23</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c01</font> iff  <a href="gate_1.html#NR1" target="_self">$</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">x1</font>,<font color="Maroon">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">x0</font>,<font color="Maroon">y1</font></span>)</span>,<a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E4:4"/><i><font color="Green">E24</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q11</font> iff  <a href="gate_1.html#NR1" target="_self">$</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">x1</font>,<font color="Maroon">y1</font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c01</font> )
 <b>and </b><br/><a NAME="E5:4"/><i><font color="Green">E25</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c11</font> iff  <a href="gate_1.html#NR1" target="_self">$</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">x1</font>,<font color="Maroon">y1</font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c01</font> )
 <b>and </b><br/><a NAME="E6:4"/><i><font color="Green">E26</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">z0</font> iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q00</font> )
 <b>and </b><br/><a NAME="E7:4"/><i><font color="Green">E27</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">z1</font> iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q01</font> )
 <b>and </b><br/><a NAME="E8:4"/><i><font color="Green">E28</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">z2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q11</font> )
 <b>and </b><br/><a NAME="E9:4"/><i><font color="Green">E29</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">z3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c11</font> )
 ;<br/>

<b>set </b><font color="Maroon">x1y0</font> =  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">x1</font>,<font color="Maroon">y0</font>;<br/>
<b>set </b><font color="Maroon">x0y1</font> =  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">x0</font>,<font color="Maroon">y1</font>;<br/>
<b>set </b><font color="Maroon">x1y1</font> =  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>;<br/>
<a NAME="E10:4"/><i><font color="Green">E33</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q11</font> iff  <a href="gate_1.html#NR1" target="_self">$</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">x1</font>,<font color="Maroon">y1</font></span>)</span>,<font color="Maroon">c01</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:4"/><i><font color="Green">E34</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c11</font> iff  <a href="gate_1.html#NR1" target="_self">$</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">x1</font>,<font color="Maroon">y1</font></span>)</span>,<font color="Maroon">c01</font> )
 
<b>by </b><i><a class="txt" href="gate_5.html#E2"><i><font color="Green">Lemma7</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E12:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">z0</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K1" target="_self">MULT210</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</font> )
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E13:4"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">z1</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K2" target="_self">MULT211</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</font> )
 <b>by </b><i><a class="txt" href="gate_5.html#E1"><i><font color="Green">Lemma4</font></i></a>, </i>;<br/>

<b>set </b><font color="Maroon">m212</font> =  <a href="gate_5.html#K3" target="_self">MULT212</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</font>;<br/>
<a NAME="E14:4"/>
 <a href="gate_5.html#K3" target="_self">MULT212</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</font> <a href="hidden.html#R1">=</a>  <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">x1</font>,<font color="Maroon">y1</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">x1</font>,<font color="Maroon">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">x0</font>,<font color="Maroon">y1</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">$</a>  <a href="gate_5.html#K3" target="_self">MULT212</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</font> iff  <a href="gate_1.html#NR1" target="_self">$</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">x1</font>,<font color="Maroon">y1</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">x1</font>,<font color="Maroon">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">x0</font>,<font color="Maroon">y1</font></span>)</span>,<a href="xboole_0.html#K1">{}</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E16:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</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">x1</font>,<font color="Maroon">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">x0</font>,<font color="Maroon">y1</font></span>)</span>,<a href="xboole_0.html#K1">{}</a>  ) or ( not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</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">x1</font>,<font color="Maroon">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">x0</font>,<font color="Maroon">y1</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">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</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">$</a> <font color="Maroon">z2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K3" target="_self">MULT212</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</font> )
 <b>by </b><i>, , , <a class="ref" href="gate_1.html#T8">GATE_1:8</a></i>;<br/>

<b>set </b><font color="Maroon">m213</font> =  <a href="gate_5.html#K4" target="_self">MULT213</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</font>;<br/>
<a NAME="E18:4"/>
 <a href="gate_5.html#K4" target="_self">MULT213</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</font> <a href="hidden.html#R1">=</a>  <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">x1</font>,<font color="Maroon">y1</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">x1</font>,<font color="Maroon">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">x0</font>,<font color="Maroon">y1</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">$</a>  <a href="gate_5.html#K4" target="_self">MULT213</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</font> iff  <a href="gate_1.html#NR1" target="_self">$</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">x1</font>,<font color="Maroon">y1</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">x1</font>,<font color="Maroon">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">x0</font>,<font color="Maroon">y1</font></span>)</span>,<a href="xboole_0.html#K1">{}</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E20:4"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K4" target="_self">MULT213</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</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">x1</font>,<font color="Maroon">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">x0</font>,<font color="Maroon">y1</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">$</a> <font color="Maroon">z3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_5.html#K4" target="_self">MULT213</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">x0</font>,<font color="Maroon">y0</font> )
 <b>by </b><i>, , , <a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>


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