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<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> be    <a href="hidden.html#M1">set</a> ;<br/>







<a NAME="E1:3"/>
( not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> iff (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>1</sub></font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>2</sub></font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E2:3"/><b>then </b><i><font color="Green">E1</font></i>: 
( not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> iff (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1">NOT1</a> <font color="Maroon">c<sub>2</sub></font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T4">GATE_1:4</a></i>;<br/>
<a NAME="E3:3"/>
(  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> iff ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>2</sub></font> ) ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T7">GATE_1:7</a></i>;<br/>
<a NAME="E4:3"/><b>then </b><i><font color="Green">E2</font></i>: 
( (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) iff (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>2</sub></font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6">GATE_1:6</a></i>;<br/>
<a NAME="E5:3"/>
(  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> iff ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>2</sub></font> ) ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T7">GATE_1:7</a></i>;<br/>
<a NAME="E6:3"/><b>then </b>
( (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) iff (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8">AND3</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>2</sub></font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T16">GATE_1:16</a></i>;<br/>
<b>hence </b><a NAME="E7:3"/>
( ( not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> implies  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) &amp; (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K1">NOT1</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> implies  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2">AND2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="gate_1.html#NR1" target="_self">$</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> implies (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) ) &amp; (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> implies  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K8">AND3</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <span class="p1">(<span class="default"><a href="gate_1.html#K8">AND3</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> implies (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>1</sub></font> implies  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>3</sub></font> ) &amp; (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>3</sub></font> implies  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K3">OR2</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="gate_2.html#E2:3"><i><font color="Green">E1</font></i></a>, <a class="txt" href="gate_2.html#E4:3"><i><font color="Green">E2</font></i></a>, <a class="ref" href="gate_1.html#T7">GATE_1:7</a></i>;<br/>


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