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

<b>let </b><font color="Maroon">c1</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">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">c4</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">p1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">sd1</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">p2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">sd2</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">p3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">sd3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">q4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">p4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">sd4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">cb1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">cb2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">l2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">t2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">l3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">m3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">t3</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">l4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">m4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">n4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">t4</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">l5</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">m5</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">n5</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">o5</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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E71</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K7" target="_self">NOR2</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> )
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green">E72</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p1</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K6" target="_self">NAND2</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> )
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green">E73</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd1</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K34" target="_self">MODADD2</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> )
 <b>and </b><br/><a NAME="E4:46"/><i><font color="Green">E74</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K7" target="_self">NOR2</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font> )
 <b>and </b><br/><a NAME="E5:46"/><i><font color="Green">E75</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K6" target="_self">NAND2</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font> )
 <b>and </b><br/><a NAME="E6:46"/><i><font color="Green">E76</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K34" target="_self">MODADD2</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font> )
 <b>and </b><br/><a NAME="E7:46"/><i><font color="Green">E77</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K7" target="_self">NOR2</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font> )
 <b>and </b><br/><a NAME="E8:46"/><i><font color="Green">E78</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K6" target="_self">NAND2</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font> )
 <b>and </b><br/><a NAME="E9:46"/><i><font color="Green">E79</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K34" target="_self">MODADD2</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font> )
 <b>and </b><br/><a NAME="E10:46"/><i><font color="Green">E80</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K7" target="_self">NOR2</a> <font color="Maroon">x4</font>,<font color="Maroon">y4</font> )
 <b>and </b><br/><a NAME="E11:46"/><i><font color="Green">E81</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K6" target="_self">NAND2</a> <font color="Maroon">x4</font>,<font color="Maroon">y4</font> )
 <b>and </b><br/><a NAME="E12:46"/><i><font color="Green">E82</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K34" target="_self">MODADD2</a> <font color="Maroon">x4</font>,<font color="Maroon">y4</font> )
 <b>and </b><br/><a NAME="E13:46"/><i><font color="Green">E122</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb1</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1" target="_self">NOT1</a> <font color="Maroon">c1</font> )
 <b>and </b><br/><a NAME="E14:46"/><i><font color="Green">E123</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K1" target="_self">NOT1</a> <font color="Maroon">cb1</font> )
 <b>and </b><br/><a NAME="E15:46"/><i><font color="Green">E124</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#K4" target="_self">XOR2</a> <font color="Maroon">cb2</font>,<font color="Maroon">sd1</font> )
 <b>and </b><br/><a NAME="E16:46"/><i><font color="Green">E125</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2" target="_self">AND2</a> <font color="Maroon">cb1</font>,<font color="Maroon">p1</font> )
 <b>and </b><br/><a NAME="E17:46"/><i><font color="Green">E126</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t2</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K7" target="_self">NOR2</a> <font color="Maroon">l2</font>,<font color="Maroon">q1</font> )
 <b>and </b><br/><a NAME="E18:46"/><i><font color="Green">E127</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#K4" target="_self">XOR2</a> <font color="Maroon">t2</font>,<font color="Maroon">sd2</font> )
 <b>and </b><br/><a NAME="E19:46"/><i><font color="Green">E128</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2" target="_self">AND2</a> <font color="Maroon">q1</font>,<font color="Maroon">p2</font> )
 <b>and </b><br/><a NAME="E20:46"/><i><font color="Green">E129</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">m3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8" target="_self">AND3</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">cb1</font> )
 <b>and </b><br/><a NAME="E21:46"/><i><font color="Green">E130</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t3</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K13" target="_self">NOR3</a> <font color="Maroon">l3</font>,<font color="Maroon">m3</font>,<font color="Maroon">q2</font> )
 <b>and </b><br/><a NAME="E22:46"/><i><font color="Green">E131</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#K4" target="_self">XOR2</a> <font color="Maroon">t3</font>,<font color="Maroon">sd3</font> )
 <b>and </b><br/><a NAME="E23:46"/><i><font color="Green">E132</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2" target="_self">AND2</a> <font color="Maroon">q2</font>,<font color="Maroon">p3</font> )
 <b>and </b><br/><a NAME="E24:46"/><i><font color="Green">E133</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">m4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8" target="_self">AND3</a> <font color="Maroon">q1</font>,<font color="Maroon">p3</font>,<font color="Maroon">p2</font> )
 <b>and </b><br/><a NAME="E25:46"/><i><font color="Green">E134</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">n4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K14" target="_self">AND4</a> <font color="Maroon">p3</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">cb1</font> )
 <b>and </b><br/><a NAME="E26:46"/><i><font color="Green">E135</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K17" target="_self">NOR4</a> <font color="Maroon">l4</font>,<font color="Maroon">m4</font>,<font color="Maroon">n4</font>,<font color="Maroon">q3</font> )
 <b>and </b><br/><a NAME="E27:46"/><i><font color="Green">E136</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#K4" target="_self">XOR2</a> <font color="Maroon">t4</font>,<font color="Maroon">sd4</font> )
 <b>and </b><br/><a NAME="E28:46"/><i><font color="Green">E137</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l5</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K2" target="_self">AND2</a> <font color="Maroon">q3</font>,<font color="Maroon">p4</font> )
 <b>and </b><br/><a NAME="E29:46"/><i><font color="Green">E138</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">m5</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K8" target="_self">AND3</a> <font color="Maroon">q2</font>,<font color="Maroon">p4</font>,<font color="Maroon">p3</font> )
 <b>and </b><br/><a NAME="E30:46"/><i><font color="Green">E139</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">n5</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K14" target="_self">AND4</a> <font color="Maroon">q1</font>,<font color="Maroon">p4</font>,<font color="Maroon">p3</font>,<font color="Maroon">p2</font> )
 <b>and </b><br/><a NAME="E31:46"/><i><font color="Green">E140</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">o5</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K18" target="_self">AND5</a> <font color="Maroon">p4</font>,<font color="Maroon">p3</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">cb1</font> )
 <b>and </b><br/><a NAME="E32:46"/><i><font color="Green">E141</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K21" target="_self">NOR5</a> <font color="Maroon">q4</font>,<font color="Maroon">l5</font>,<font color="Maroon">m5</font>,<font color="Maroon">n5</font>,<font color="Maroon">o5</font> )
 ;<br/>

<a NAME="E33:46"/><i><font color="Green">E142</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#D22" target="_self">Def22</a>, , </i>;<br/>
<a NAME="E34:46"/><i><font color="Green">E143</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> iff  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> )
 
<b>by </b><i><a class="ref" href="gate_1.html#D23" target="_self">Def23</a>, , </i>;<br/>
<a NAME="E35:46"/><i><font color="Green">E144</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd1</font> iff ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> ) &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> ) ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#D24" target="_self">Def24</a>, <a class="ref" href="gate_1.html#D21" target="_self">Def21</a>, </i>;<br/>
<a NAME="E36:46"/><i><font color="Green">E145</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#D25" target="_self">Def25</a>, , </i>;<br/>
<a NAME="E37:46"/><i><font color="Green">E146</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> iff  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> )
 
<b>by </b><i><a class="ref" href="gate_1.html#D26" target="_self">Def26</a>, , </i>;<br/>
<a NAME="E38:46"/><i><font color="Green">E147</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd2</font> iff ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> ) &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> ) ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#D27" target="_self">Def27</a>, <a class="ref" href="gate_1.html#D21" target="_self">Def21</a>, </i>;<br/>
<a NAME="E39:46"/><i><font color="Green">E148</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x3</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y3</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#D28" target="_self">Def28</a>, , </i>;<br/>
<a NAME="E40:46"/><i><font color="Green">E149</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x3</font> iff  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y3</font> )
 
<b>by </b><i><a class="ref" href="gate_1.html#D29" target="_self">Def29</a>, , </i>;<br/>
<a NAME="E41:46"/><i><font color="Green">E150</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd3</font> iff ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x3</font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y3</font> ) &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x3</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y3</font> ) ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#D30" target="_self">Def30</a>, <a class="ref" href="gate_1.html#D21" target="_self">Def21</a>, </i>;<br/>
<a NAME="E42:46"/><i><font color="Green">E151</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q4</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#D31" target="_self">Def31</a>, , </i>;<br/>
<a NAME="E43:46"/><i><font color="Green">E152</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> iff  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> )
 
<b>by </b><i><a class="ref" href="gate_1.html#D32" target="_self">Def32</a>, , </i>;<br/>
<a NAME="E44:46"/><i><font color="Green">E153</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd4</font> iff ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#D33" target="_self">Def33</a>, <a class="ref" href="gate_1.html#D21" target="_self">Def21</a>, </i>;<br/>
<a NAME="E45:46"/><i><font color="Green">E154</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb1</font> iff not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E46:46"/><i><font color="Green">E155</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb2</font> iff not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E47:46"/><i><font color="Green">E156</font></i>: 
( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd1</font> ) or ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd1</font> ) ) iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s1</font> )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E48:46"/><i><font color="Green">E157</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l2</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p1</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E49:46"/><i><font color="Green">E158</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t2</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E50:46"/><i><font color="Green">E159</font></i>: 
( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t2</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd2</font> ) or ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd2</font> ) ) iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s2</font> )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E51:46"/><i><font color="Green">E160</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l3</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E52:46"/><i><font color="Green">E161</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">m3</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb1</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E53:46"/><i><font color="Green">E162</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t3</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l3</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">m3</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> ) )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#D3" target="_self">Def3</a>, </i>;<br/>
<a NAME="E54:46"/><i><font color="Green">E163</font></i>: 
( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t3</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd3</font> ) or ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd3</font> ) ) iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s3</font> )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E55:46"/><i><font color="Green">E164</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l4</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E56:46"/><i><font color="Green">E165</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">m4</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E57:46"/><i><font color="Green">E166</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">n4</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb1</font> ) )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E58:46"/><i><font color="Green">E167</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t4</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">m4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">n4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> ) )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#D6" target="_self">Def6</a>, </i>;<br/>
<a NAME="E59:46"/><i><font color="Green">E168</font></i>: 
( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd4</font> ) or ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">t4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">sd4</font> ) ) iff  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">s4</font> )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E60:46"/><i><font color="Green">E169</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l5</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p4</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E61:46"/><i><font color="Green">E170</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">m5</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E62:46"/><i><font color="Green">E171</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">n5</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> ) )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E63:46"/><i><font color="Green">E172</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">o5</font> iff (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb1</font> ) )
 
<b>by </b><i>, <a class="ref" href="gate_1.html#D7" target="_self">Def7</a>, </i>;<br/>
<a NAME="E64:46"/><i><font color="Green">E173</font></i>: 
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c4</font> iff ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">l5</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">m5</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">n5</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">o5</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E1:46"><i><font color="Green">E71</font></i></a>, <a class="ref" href="gate_1.html#D10" target="_self">Def10</a>, </i>;<br/>
<b>thus </b><a NAME="E65:46"/>
(  <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#NK35" target="_self">ADD1</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> )
 <b>by </b><i><a class="txt" href="gate_1.html#E4:46"><i><font color="Green">E74</font></i></a>, , , , </i>;<br/>

<a NAME="E66:46"/><i><font color="Green">E174</font></i>: 
( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> ) ) iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K11" target="_self">MAJ3</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E67:46"/><b>then </b>
(  <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#K10" target="_self">XOR3</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#NK36" target="_self">CARR1</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E2:46"><i><font color="Green">E72</font></i></a>, <a class="txt" href="gate_1.html#E3:46"><i><font color="Green">E73</font></i></a>, <a class="txt" href="gate_1.html#E7:46"><i><font color="Green">E77</font></i></a>, , , , , </i>;<br/>
<b>hence </b><a NAME="E68:46"/>
(  <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#K35" target="_self">ADD2</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> )
 ;<br/>

<a NAME="E69:46"/><i><font color="Green">E175</font></i>: 
( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#NK36" target="_self">CARR1</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#NK36" target="_self">CARR1</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> ) ) iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K11" target="_self">MAJ3</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#NK36" target="_self">CARR1</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E70:46"/><b>then </b><i><font color="Green">E176</font></i>: 
( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> &amp; ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> ) ) ) or ( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> ) ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> ) ) iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K36" target="_self">CARR2</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E71:46"/><b>then </b>
(  <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#K10" target="_self">XOR3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K36" target="_self">CARR2</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E2:46"><i><font color="Green">E72</font></i></a>, <a class="txt" href="gate_1.html#E3:46"><i><font color="Green">E73</font></i></a>, <a class="txt" href="gate_1.html#E5:46"><i><font color="Green">E75</font></i></a>, <a class="txt" href="gate_1.html#E6:46"><i><font color="Green">E76</font></i></a>, <a class="txt" href="gate_1.html#E10:46"><i><font color="Green">E80</font></i></a>, , , , , , </i>;<br/>
<b>hence </b><a NAME="E72:46"/>
(  <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#K37" target="_self">ADD3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> )
 ;<br/>

<a NAME="E73:46"/><i><font color="Green">E177</font></i>: 
( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y3</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K36" target="_self">CARR2</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K36" target="_self">CARR2</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x3</font> ) ) iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K11" target="_self">MAJ3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K36" target="_self">CARR2</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">b</font> =  <a href="gate_1.html#K38" target="_self">CARR3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font>;<br/>
<a NAME="E74:46"/><i><font color="Green">E178</font></i>: 
( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y3</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y3</font> &amp; ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> &amp; ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> ) ) ) or ( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> ) ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> ) ) ) or ( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y2</font> &amp; ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> ) ) ) or ( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x1</font> ) ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x2</font> ) ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x3</font> ) ) iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K38" target="_self">CARR3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E75:46"/>
( <!--HUMANRECFAILED-->( ( not <!--HUMANRECFAILED-->( not <!--HUMANRECFAILED-->(  <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">p3</font> ) &amp; not <!--HUMANRECFAILED-->(  <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">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p1</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp; ( ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) ) ) &amp; not <!--HUMANRECFAILED-->( ( (  <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">p3</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">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p1</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> ) or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> ) &amp; (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) ) ) or ( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) or ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K38" target="_self">CARR3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K38" target="_self">CARR3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> ) ) &amp; ( ( not <!--HUMANRECFAILED-->( ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) or ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K38" target="_self">CARR3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> ) &amp; not <!--HUMANRECFAILED-->( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K38" target="_self">CARR3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> ) ) or ( not <!--HUMANRECFAILED-->(  <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">p3</font> ) &amp; not <!--HUMANRECFAILED-->(  <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">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p1</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> ) &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp; ( ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) ) ) or ( ( (  <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">p3</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">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p1</font> &amp; not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c1</font> ) or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> ) &amp; (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> or  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) &amp; (  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> or  not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) ) ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E2:46"><i><font color="Green">E72</font></i></a>, <a class="txt" href="gate_1.html#E3:46"><i><font color="Green">E73</font></i></a>, <a class="txt" href="gate_1.html#E5:46"><i><font color="Green">E75</font></i></a>, <a class="txt" href="gate_1.html#E6:46"><i><font color="Green">E76</font></i></a>, <a class="txt" href="gate_1.html#E8:46"><i><font color="Green">E78</font></i></a>, <a class="txt" href="gate_1.html#E9:46"><i><font color="Green">E79</font></i></a>, , </i>;<br/>
<a NAME="E76:46"/><b>then </b>
(  <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#K10" target="_self">XOR3</a> <font color="Maroon">x4</font>,<font color="Maroon">y4</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K38" target="_self">CARR3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#D34" target="_self">Def34</a>, , , , , , , </i>;<br/>
<b>hence </b><a NAME="E77:46"/>
(  <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#K39" target="_self">ADD4</a> <font color="Maroon">x4</font>,<font color="Maroon">y4</font>,<font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> )
 ;<br/>

<a NAME="E78:46"/>
( ( not  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q4</font> &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p4</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">q2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> ) &amp; not <!--HUMANRECFAILED-->(  <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">p4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p3</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p2</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">p1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">cb1</font> ) ) iff ( (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">y4</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K38" target="_self">CARR3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> ) or (  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K38" target="_self">CARR3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> &amp;  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">x4</font> ) ) )
 
<b>by </b><i>, <a class="txt" href="gate_1.html#E2:46"><i><font color="Green">E72</font></i></a>, <a class="txt" href="gate_1.html#E3:46"><i><font color="Green">E73</font></i></a>, <a class="txt" href="gate_1.html#E5:46"><i><font color="Green">E75</font></i></a>, <a class="txt" href="gate_1.html#E6:46"><i><font color="Green">E76</font></i></a>, <a class="txt" href="gate_1.html#E8:46"><i><font color="Green">E78</font></i></a>, <a class="txt" href="gate_1.html#E9:46"><i><font color="Green">E79</font></i></a>, <a class="txt" href="gate_1.html#E11:46"><i><font color="Green">E81</font></i></a>, <a class="txt" href="gate_1.html#E12:46"><i><font color="Green">E82</font></i></a>, , </i>;<br/>
<a NAME="E79:46"/><b>then </b>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K11" target="_self">MAJ3</a> <font color="Maroon">x4</font>,<font color="Maroon">y4</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K38" target="_self">CARR3</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font></span>)</span> )
 
<b>by </b><i>, , , , , <a class="ref" href="gate_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E80:46"/>
(  <a href="gate_1.html#NR1" target="_self">$</a> <font color="Maroon">c4</font> iff  <a href="gate_1.html#NR1" target="_self">$</a>  <a href="gate_1.html#K40" target="_self">CARR4</a> <font color="Maroon">x4</font>,<font color="Maroon">y4</font>,<font color="Maroon">x3</font>,<font color="Maroon">y3</font>,<font color="Maroon">x2</font>,<font color="Maroon">y2</font>,<font color="Maroon">x1</font>,<font color="Maroon">y1</font>,<font color="Maroon">c1</font> )
 ;<br/>


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