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

<b>let </b><font color="Maroon" title="c1">c1</font>, <font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">x2</font>, <font color="Maroon" title="c5">y2</font>, <font color="Maroon" title="c6">x3</font>, <font color="Maroon" title="c7">y3</font>, <font color="Maroon" title="c8">x4</font>, <font color="Maroon" title="c9">y4</font>, <font color="Maroon" title="c10">c4</font>, <font color="Maroon" title="c11">q1</font>, <font color="Maroon" title="c12">p1</font>, <font color="Maroon" title="c13">sd1</font>, <font color="Maroon" title="c14">q2</font>, <font color="Maroon" title="c15">p2</font>, <font color="Maroon" title="c16">sd2</font>, <font color="Maroon" title="c17">q3</font>, <font color="Maroon" title="c18">p3</font>, <font color="Maroon" title="c19">sd3</font>, <font color="Maroon" title="c20">q4</font>, <font color="Maroon" title="c21">p4</font>, <font color="Maroon" title="c22">sd4</font>, <font color="Maroon" title="c23">cb1</font>, <font color="Maroon" title="c24">cb2</font>, <font color="Maroon" title="c25">l2</font>, <font color="Maroon" title="c26">t2</font>, <font color="Maroon" title="c27">l3</font>, <font color="Maroon" title="c28">m3</font>, <font color="Maroon" title="c29">t3</font>, <font color="Maroon" title="c30">l4</font>, <font color="Maroon" title="c31">m4</font>, <font color="Maroon" title="c32">n4</font>, <font color="Maroon" title="c33">t4</font>, <font color="Maroon" title="c34">l5</font>, <font color="Maroon" title="c35">m5</font>, <font color="Maroon" title="c36">n5</font>, <font color="Maroon" title="c37">o5</font>, <font color="Maroon" title="c38">s1</font>, <font color="Maroon" title="c39">s2</font>, <font color="Maroon" title="c40">s3</font>, <font color="Maroon" title="c41">s4</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <!--HUMANRECFAILED-->( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">sd1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">sd1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">sd2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">sd2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">q3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">q3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">sd3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">sd3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">q4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">q4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">sd4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">sd4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">cb1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">cb1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">cb2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">cb1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">cb1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">cb2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">s1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">cb2</font>,<font color="Maroon" title="c13">sd1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">cb2</font>,<font color="Maroon" title="c13">sd1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">s1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">l2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c23">cb1</font>,<font color="Maroon" title="c12">p1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c23">cb1</font>,<font color="Maroon" title="c12">p1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">l2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">t2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c25">l2</font>,<font color="Maroon" title="c11">q1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c25">l2</font>,<font color="Maroon" title="c11">q1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">t2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">t2</font>,<font color="Maroon" title="c16">sd2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">t2</font>,<font color="Maroon" title="c16">sd2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">l3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">q1</font>,<font color="Maroon" title="c15">p2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">q1</font>,<font color="Maroon" title="c15">p2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">l3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">m3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c23">cb1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c23">cb1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">m3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">t3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K13" title="GATE_1:func.13">NOR3</a> <font color="Maroon" title="c27">l3</font>,<font color="Maroon" title="c28">m3</font>,<font color="Maroon" title="c14">q2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K13" title="GATE_1:func.13">NOR3</a> <font color="Maroon" title="c27">l3</font>,<font color="Maroon" title="c28">m3</font>,<font color="Maroon" title="c14">q2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">t3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">t3</font>,<font color="Maroon" title="c19">sd3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">t3</font>,<font color="Maroon" title="c19">sd3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">l4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">q2</font>,<font color="Maroon" title="c18">p3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">q2</font>,<font color="Maroon" title="c18">p3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">l4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">m4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c11">q1</font>,<font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c11">q1</font>,<font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">m4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">n4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K14" title="GATE_1:func.14">AND4</a> <font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c23">cb1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K14" title="GATE_1:func.14">AND4</a> <font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c23">cb1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">n4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">t4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K17" title="GATE_1:func.17">NOR4</a> <font color="Maroon" title="c30">l4</font>,<font color="Maroon" title="c31">m4</font>,<font color="Maroon" title="c32">n4</font>,<font color="Maroon" title="c17">q3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K17" title="GATE_1:func.17">NOR4</a> <font color="Maroon" title="c30">l4</font>,<font color="Maroon" title="c31">m4</font>,<font color="Maroon" title="c32">n4</font>,<font color="Maroon" title="c17">q3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">t4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c33">t4</font>,<font color="Maroon" title="c22">sd4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c33">t4</font>,<font color="Maroon" title="c22">sd4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">l5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q3</font>,<font color="Maroon" title="c21">p4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q3</font>,<font color="Maroon" title="c21">p4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">l5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">m5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c14">q2</font>,<font color="Maroon" title="c21">p4</font>,<font color="Maroon" title="c18">p3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c14">q2</font>,<font color="Maroon" title="c21">p4</font>,<font color="Maroon" title="c18">p3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">m5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">n5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K14" title="GATE_1:func.14">AND4</a> <font color="Maroon" title="c11">q1</font>,<font color="Maroon" title="c21">p4</font>,<font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K14" title="GATE_1:func.14">AND4</a> <font color="Maroon" title="c11">q1</font>,<font color="Maroon" title="c21">p4</font>,<font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">n5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">o5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K18" title="GATE_1:func.18">AND5</a> <font color="Maroon" title="c21">p4</font>,<font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c23">cb1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K18" title="GATE_1:func.18">AND5</a> <font color="Maroon" title="c21">p4</font>,<font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c23">cb1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">o5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K21" title="GATE_1:func.21">NOR5</a> <font color="Maroon" title="c20">q4</font>,<font color="Maroon" title="c34">l5</font>,<font color="Maroon" title="c35">m5</font>,<font color="Maroon" title="c36">n5</font>,<font color="Maroon" title="c37">o5</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K21" title="GATE_1:func.21">NOR5</a> <font color="Maroon" title="c20">q4</font>,<font color="Maroon" title="c34">l5</font>,<font color="Maroon" title="c35">m5</font>,<font color="Maroon" title="c36">n5</font>,<font color="Maroon" title="c37">o5</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> ) &amp; not <!--HUMANRECFAILED-->( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">s1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#NK35" title="GATE_1:NK.35">ADD1</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#NK35" title="GATE_1:NK.35">ADD1</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">s1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K35" title="GATE_1:func.35">ADD2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K35" title="GATE_1:func.35">ADD2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K37" title="GATE_1:func.37">ADD3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K37" title="GATE_1:func.37">ADD3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K39" title="GATE_1:func.39">ADD4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K39" title="GATE_1:func.39">ADD4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> ) ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green" title="E40">A1</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font> )
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green" title="E41">A2</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font> )
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green" title="E42">A3</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">sd1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font> )
 <b>and </b><br/><a NAME="E4:46"/><i><font color="Green" title="E43">A4</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font> )
 <b>and </b><br/><a NAME="E5:46"/><i><font color="Green" title="E44">A5</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font> )
 <b>and </b><br/><a NAME="E6:46"/><i><font color="Green" title="E45">A6</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">sd2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font> )
 <b>and </b><br/><a NAME="E7:46"/><i><font color="Green" title="E46">A7</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">q3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font> )
 <b>and </b><br/><a NAME="E8:46"/><i><font color="Green" title="E47">A8</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font> )
 <b>and </b><br/><a NAME="E9:46"/><i><font color="Green" title="E48">A9</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">sd3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font> )
 <b>and </b><br/><a NAME="E10:46"/><i><font color="Green" title="E49">A10</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">q4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font> )
 <b>and </b><br/><a NAME="E11:46"/><i><font color="Green" title="E50">A11</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K6" title="GATE_1:func.6">NAND2</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font> )
 <b>and </b><br/><a NAME="E12:46"/><i><font color="Green" title="E51">A12</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">sd4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K34" title="GATE_1:func.34">MODADD2</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font> )
 <b>and </b><br/><a NAME="E13:46"/><i><font color="Green" title="E52">A13</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">cb1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c1">c1</font> )
 <b>and </b><br/><a NAME="E14:46"/><i><font color="Green" title="E53">A14</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">cb2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K1" title="GATE_1:func.1">NOT1</a> <font color="Maroon" title="c23">cb1</font> )
 <b>and </b><br/><a NAME="E15:46"/><i><font color="Green" title="E54">A15</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">s1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c24">cb2</font>,<font color="Maroon" title="c13">sd1</font> )
 <b>and </b><br/><a NAME="E16:46"/><i><font color="Green" title="E55">A16</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">l2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c23">cb1</font>,<font color="Maroon" title="c12">p1</font> )
 <b>and </b><br/><a NAME="E17:46"/><i><font color="Green" title="E56">A17</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">t2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K7" title="GATE_1:func.7">NOR2</a> <font color="Maroon" title="c25">l2</font>,<font color="Maroon" title="c11">q1</font> )
 <b>and </b><br/><a NAME="E18:46"/><i><font color="Green" title="E57">A18</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c26">t2</font>,<font color="Maroon" title="c16">sd2</font> )
 <b>and </b><br/><a NAME="E19:46"/><i><font color="Green" title="E58">A19</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">l3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c11">q1</font>,<font color="Maroon" title="c15">p2</font> )
 <b>and </b><br/><a NAME="E20:46"/><i><font color="Green" title="E59">A20</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">m3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c23">cb1</font> )
 <b>and </b><br/><a NAME="E21:46"/><i><font color="Green" title="E60">A21</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">t3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K13" title="GATE_1:func.13">NOR3</a> <font color="Maroon" title="c27">l3</font>,<font color="Maroon" title="c28">m3</font>,<font color="Maroon" title="c14">q2</font> )
 <b>and </b><br/><a NAME="E22:46"/><i><font color="Green" title="E61">A22</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c29">t3</font>,<font color="Maroon" title="c19">sd3</font> )
 <b>and </b><br/><a NAME="E23:46"/><i><font color="Green" title="E62">A23</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">l4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c14">q2</font>,<font color="Maroon" title="c18">p3</font> )
 <b>and </b><br/><a NAME="E24:46"/><i><font color="Green" title="E63">A24</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">m4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c11">q1</font>,<font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font> )
 <b>and </b><br/><a NAME="E25:46"/><i><font color="Green" title="E64">A25</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">n4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K14" title="GATE_1:func.14">AND4</a> <font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c23">cb1</font> )
 <b>and </b><br/><a NAME="E26:46"/><i><font color="Green" title="E65">A26</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">t4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K17" title="GATE_1:func.17">NOR4</a> <font color="Maroon" title="c30">l4</font>,<font color="Maroon" title="c31">m4</font>,<font color="Maroon" title="c32">n4</font>,<font color="Maroon" title="c17">q3</font> )
 <b>and </b><br/><a NAME="E27:46"/><i><font color="Green" title="E66">A27</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c33">t4</font>,<font color="Maroon" title="c22">sd4</font> )
 <b>and </b><br/><a NAME="E28:46"/><i><font color="Green" title="E67">A28</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">l5</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c17">q3</font>,<font color="Maroon" title="c21">p4</font> )
 <b>and </b><br/><a NAME="E29:46"/><i><font color="Green" title="E68">A29</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">m5</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K8" title="GATE_1:func.8">AND3</a> <font color="Maroon" title="c14">q2</font>,<font color="Maroon" title="c21">p4</font>,<font color="Maroon" title="c18">p3</font> )
 <b>and </b><br/><a NAME="E30:46"/><i><font color="Green" title="E69">A30</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">n5</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K14" title="GATE_1:func.14">AND4</a> <font color="Maroon" title="c11">q1</font>,<font color="Maroon" title="c21">p4</font>,<font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font> )
 <b>and </b><br/><a NAME="E31:46"/><i><font color="Green" title="E70">A31</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">o5</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K18" title="GATE_1:func.18">AND5</a> <font color="Maroon" title="c21">p4</font>,<font color="Maroon" title="c18">p3</font>,<font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c23">cb1</font> )
 <b>and </b><br/><a NAME="E32:46"/><i><font color="Green" title="E71">A32</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K21" title="GATE_1:func.21">NOR5</a> <font color="Maroon" title="c20">q4</font>,<font color="Maroon" title="c34">l5</font>,<font color="Maroon" title="c35">m5</font>,<font color="Maroon" title="c36">n5</font>,<font color="Maroon" title="c37">o5</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">s1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#NK35" title="GATE_1:NK.35">ADD1</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#NK35" title="GATE_1:NK.35">ADD1</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">s1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K35" title="GATE_1:func.35">ADD2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K35" title="GATE_1:func.35">ADD2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K37" title="GATE_1:func.37">ADD3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K37" title="GATE_1:func.37">ADD3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K39" title="GATE_1:func.39">ADD4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K39" title="GATE_1:func.39">ADD4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> ) )</span><br/>

<a NAME="E33:46"/><i><font color="Green" title="E72">A33</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> iff ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E1:46"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="gate_1.html#D7" target="_self" title="GATE_1:def.7">Def7</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E34:46"/><i><font color="Green" title="E73">A34</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> iff  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E2:46"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="gate_1.html#D6" target="_self" title="GATE_1:def.6">Def6</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E35:46"/><i><font color="Green" title="E74">A35</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">sd1</font> iff ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> ) ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E3:46"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="gate_1.html#D34" target="_self" title="GATE_1:def.34">Def34</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E36:46"/><i><font color="Green" title="E75">A36</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> iff ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E4:46"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="gate_1.html#D7" target="_self" title="GATE_1:def.7">Def7</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E37:46"/><i><font color="Green" title="E76">A37</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> iff  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E5:46"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="gate_1.html#D6" target="_self" title="GATE_1:def.6">Def6</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E38:46"/><i><font color="Green" title="E77">A38</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">sd2</font> iff ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> ) ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E6:46"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="gate_1.html#D34" target="_self" title="GATE_1:def.34">Def34</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E39:46"/><i><font color="Green" title="E78">A39</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">q3</font> iff ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">x3</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">y3</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E7:46"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="gate_1.html#D7" target="_self" title="GATE_1:def.7">Def7</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E40:46"/><i><font color="Green" title="E79">A40</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">x3</font> iff  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">y3</font> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E8:46"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="gate_1.html#D6" target="_self" title="GATE_1:def.6">Def6</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E41:46"/><i><font color="Green" title="E80">A41</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">sd3</font> iff ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">x3</font> or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">y3</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">x3</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">y3</font> ) ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E9:46"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="gate_1.html#D34" target="_self" title="GATE_1:def.34">Def34</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E42:46"/><i><font color="Green" title="E81">A42</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">q4</font> iff ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E10:46"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="gate_1.html#D7" target="_self" title="GATE_1:def.7">Def7</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E43:46"/><i><font color="Green" title="E82">A43</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> iff  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E11:46"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="gate_1.html#D6" target="_self" title="GATE_1:def.6">Def6</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E44:46"/><i><font color="Green" title="E83">A44</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">sd4</font> iff ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E12:46"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="gate_1.html#D34" target="_self" title="GATE_1:def.34">Def34</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E45:46"/><i><font color="Green" title="E84">A45</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">cb1</font> iff not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E13:46"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="gate_1.html#T4" target="_self" title="GATE_1:th.4">Th4</a></i>;<br/>
<a NAME="E46:46"/><i><font color="Green" title="E85">A46</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">cb2</font> iff not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">cb1</font> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E14:46"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="gate_1.html#T4" target="_self" title="GATE_1:th.4">Th4</a></i>;<br/>
<a NAME="E47:46"/><i><font color="Green" title="E86">A47</font></i>: 
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">cb2</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">sd1</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c24">cb2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">sd1</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">s1</font> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E15:46"><i><font color="Green" title="E54">A15</font></i></a>, <a class="ref" href="gate_1.html#D4" target="_self" title="GATE_1:def.4">Def4</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E48:46"/><i><font color="Green" title="E87">A48</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">l2</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">cb1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E16:46"><i><font color="Green" title="E55">A16</font></i></a>, <a class="ref" href="gate_1.html#D2" target="_self" title="GATE_1:def.2">Def2</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E49:46"/><i><font color="Green" title="E88">A49</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">t2</font> iff ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c25">l2</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E17:46"><i><font color="Green" title="E56">A17</font></i></a>, <a class="ref" href="gate_1.html#D7" target="_self" title="GATE_1:def.7">Def7</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E50:46"/><i><font color="Green" title="E89">A50</font></i>: 
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">t2</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">sd2</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c26">t2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c16">sd2</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E18:46"><i><font color="Green" title="E57">A18</font></i></a>, <a class="ref" href="gate_1.html#D4" target="_self" title="GATE_1:def.4">Def4</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E51:46"/><i><font color="Green" title="E90">A51</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">l3</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E19:46"><i><font color="Green" title="E58">A19</font></i></a>, <a class="ref" href="gate_1.html#D2" target="_self" title="GATE_1:def.2">Def2</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E52:46"/><i><font color="Green" title="E91">A52</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">m3</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">cb1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E20:46"><i><font color="Green" title="E59">A20</font></i></a>, <a class="ref" href="gate_1.html#D8" target="_self" title="GATE_1:def.8">Def8</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E53:46"/><i><font color="Green" title="E92">A53</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">t3</font> iff ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c27">l3</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c28">m3</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E21:46"><i><font color="Green" title="E60">A21</font></i></a>, <a class="ref" href="gate_1.html#D13" target="_self" title="GATE_1:def.13">Def13</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E54:46"/><i><font color="Green" title="E93">A54</font></i>: 
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">t3</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">sd3</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c29">t3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c19">sd3</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E22:46"><i><font color="Green" title="E61">A22</font></i></a>, <a class="ref" href="gate_1.html#D4" target="_self" title="GATE_1:def.4">Def4</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E55:46"/><i><font color="Green" title="E94">A55</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">l4</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E23:46"><i><font color="Green" title="E62">A23</font></i></a>, <a class="ref" href="gate_1.html#D2" target="_self" title="GATE_1:def.2">Def2</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E56:46"/><i><font color="Green" title="E95">A56</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">m4</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E24:46"><i><font color="Green" title="E63">A24</font></i></a>, <a class="ref" href="gate_1.html#D8" target="_self" title="GATE_1:def.8">Def8</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E57:46"/><i><font color="Green" title="E96">A57</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">n4</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">cb1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E25:46"><i><font color="Green" title="E64">A25</font></i></a>, <a class="ref" href="gate_1.html#D14" target="_self" title="GATE_1:def.14">Def14</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E58:46"/><i><font color="Green" title="E97">A58</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">t4</font> iff ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c30">l4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c31">m4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c32">n4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">q3</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E26:46"><i><font color="Green" title="E65">A26</font></i></a>, <a class="ref" href="gate_1.html#D17" target="_self" title="GATE_1:def.17">Def17</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E59:46"/><i><font color="Green" title="E98">A59</font></i>: 
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">t4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">sd4</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c33">t4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c22">sd4</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E27:46"><i><font color="Green" title="E66">A27</font></i></a>, <a class="ref" href="gate_1.html#D4" target="_self" title="GATE_1:def.4">Def4</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E60:46"/><i><font color="Green" title="E99">A60</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">l5</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">q3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E28:46"><i><font color="Green" title="E67">A28</font></i></a>, <a class="ref" href="gate_1.html#D2" target="_self" title="GATE_1:def.2">Def2</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E61:46"/><i><font color="Green" title="E100">A61</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">m5</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E29:46"><i><font color="Green" title="E68">A29</font></i></a>, <a class="ref" href="gate_1.html#D8" target="_self" title="GATE_1:def.8">Def8</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E62:46"/><i><font color="Green" title="E101">A62</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">n5</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E30:46"><i><font color="Green" title="E69">A30</font></i></a>, <a class="ref" href="gate_1.html#D14" target="_self" title="GATE_1:def.14">Def14</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E63:46"/><i><font color="Green" title="E102">A63</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">o5</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">cb1</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E31:46"><i><font color="Green" title="E70">A31</font></i></a>, <a class="ref" href="gate_1.html#D18" target="_self" title="GATE_1:def.18">Def18</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E64:46"/><i><font color="Green" title="E103">A64</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> iff ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">q4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c34">l5</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c35">m5</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c36">n5</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c37">o5</font> ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E32:46"><i><font color="Green" title="E71">A32</font></i></a>, <a class="ref" href="gate_1.html#D21" target="_self" title="GATE_1:def.21">Def21</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<b>thus </b><a NAME="E65:46"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c38">s1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#NK35" title="GATE_1:NK.35">ADD1</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> )
 <b>by </b><i><a class="txt" href="gate_1.html#E35:46"><i><font color="Green" title="E74">A35</font></i></a>, <a class="txt" href="gate_1.html#E45:46"><i><font color="Green" title="E84">A45</font></i></a>, <a class="txt" href="gate_1.html#E46:46"><i><font color="Green" title="E85">A46</font></i></a>, <a class="txt" href="gate_1.html#E47:46"><i><font color="Green" title="E86">A47</font></i></a>, <a class="ref" href="gate_1.html#T18" target="_self" title="GATE_1:th.18">Th18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K35" title="GATE_1:func.35">ADD2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K35" title="GATE_1:func.35">ADD2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K37" title="GATE_1:func.37">ADD3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K37" title="GATE_1:func.37">ADD3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K39" title="GATE_1:func.39">ADD4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K39" title="GATE_1:func.39">ADD4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> ) )</span><br/>

<a NAME="E66:46"/><i><font color="Green" title="E104">A65</font></i>: 
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> )
 
<b>by </b><i><a class="ref" href="gate_1.html#D11" target="_self" title="GATE_1:def.11">Def11</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E67:46"/><b>then </b>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#NK36" title="GATE_1:NK.36">CARR1</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E33:46"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="gate_1.html#E34:46"><i><font color="Green" title="E73">A34</font></i></a>, <a class="txt" href="gate_1.html#E38:46"><i><font color="Green" title="E77">A38</font></i></a>, <a class="txt" href="gate_1.html#E45:46"><i><font color="Green" title="E84">A45</font></i></a>, <a class="txt" href="gate_1.html#E48:46"><i><font color="Green" title="E87">A48</font></i></a>, <a class="txt" href="gate_1.html#E49:46"><i><font color="Green" title="E88">A49</font></i></a>, <a class="txt" href="gate_1.html#E50:46"><i><font color="Green" title="E89">A50</font></i></a>, <a class="ref" href="gate_1.html#T18" target="_self" title="GATE_1:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E68:46"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c39">s2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K35" title="GATE_1:func.35">ADD2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K37" title="GATE_1:func.37">ADD3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K37" title="GATE_1:func.37">ADD3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K39" title="GATE_1:func.39">ADD4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K39" title="GATE_1:func.39">ADD4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> ) )</span><br/>

<a NAME="E69:46"/><i><font color="Green" title="E105">A66</font></i>: 
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#NK36" title="GATE_1:NK.36">CARR1</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#NK36" title="GATE_1:NK.36">CARR1</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<span class="p1">(<span class="default"><a href="gate_1.html#NK36" title="GATE_1:NK.36">CARR1</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#D11" target="_self" title="GATE_1:def.11">Def11</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<a NAME="E70:46"/><b>then </b><i><font color="Green" title="E106">A67</font></i>: 
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> &amp; ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> ) ) ) or ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> ) ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K36" title="GATE_1:func.36">CARR2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T19" target="_self" title="GATE_1:th.19">Th19</a></i>;<br/>
<a NAME="E71:46"/><b>then </b>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K36" title="GATE_1:func.36">CARR2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E33:46"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="gate_1.html#E34:46"><i><font color="Green" title="E73">A34</font></i></a>, <a class="txt" href="gate_1.html#E36:46"><i><font color="Green" title="E75">A36</font></i></a>, <a class="txt" href="gate_1.html#E37:46"><i><font color="Green" title="E76">A37</font></i></a>, <a class="txt" href="gate_1.html#E41:46"><i><font color="Green" title="E80">A41</font></i></a>, <a class="txt" href="gate_1.html#E45:46"><i><font color="Green" title="E84">A45</font></i></a>, <a class="txt" href="gate_1.html#E51:46"><i><font color="Green" title="E90">A51</font></i></a>, <a class="txt" href="gate_1.html#E52:46"><i><font color="Green" title="E91">A52</font></i></a>, <a class="txt" href="gate_1.html#E53:46"><i><font color="Green" title="E92">A53</font></i></a>, <a class="txt" href="gate_1.html#E54:46"><i><font color="Green" title="E93">A54</font></i></a>, <a class="ref" href="gate_1.html#T18" target="_self" title="GATE_1:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E72:46"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c40">s3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K37" title="GATE_1:func.37">ADD3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K39" title="GATE_1:func.39">ADD4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K39" title="GATE_1:func.39">ADD4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> ) )</span><br/>

<a NAME="E73:46"/><i><font color="Green" title="E107">A68</font></i>: 
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">x3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">y3</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">y3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K36" title="GATE_1:func.36">CARR2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K36" title="GATE_1:func.36">CARR2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">x3</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K36" title="GATE_1:func.36">CARR2</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="gate_1.html#D11" target="_self" title="GATE_1:def.11">Def11</a>, <a class="ref" href="gate_1.html#T5" target="_self" title="GATE_1:th.5">Th5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c42">b</font> =  <a href="gate_1.html#K38" title="GATE_1:func.38">CARR3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font>;<br/>
<a NAME="E74:46"/><i><font color="Green" title="E108">A69</font></i>: 
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">x3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">y3</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">y3</font> &amp; ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> &amp; ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> ) ) ) or ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> ) ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> ) ) ) or ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">y2</font> &amp; ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> ) ) ) or ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c3">y1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c2">x1</font> ) ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c4">x2</font> ) ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">x3</font> ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K38" title="GATE_1:func.38">CARR3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E66:46"><i><font color="Green" title="E104">A65</font></i></a>, <a class="txt" href="gate_1.html#E69:46"><i><font color="Green" title="E105">A66</font></i></a>, <a class="txt" href="gate_1.html#E73:46"><i><font color="Green" title="E107">A68</font></i></a></i>;<br/>
<a NAME="E75:46"/>
( <!--HUMANRECFAILED-->( ( not <!--HUMANRECFAILED-->( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">q3</font> &amp; ( ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) ) ) &amp; not <!--HUMANRECFAILED-->( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> ) or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">q3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) ) ) or ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K38" title="GATE_1:func.38">CARR3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K38" title="GATE_1:func.38">CARR3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) ) &amp; ( ( not <!--HUMANRECFAILED-->( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K38" title="GATE_1:func.38">CARR3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) &amp; not <!--HUMANRECFAILED-->( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K38" title="GATE_1:func.38">CARR3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) ) or ( not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> ) &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">q3</font> &amp; ( ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) ) ) or ( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c1">c1</font> ) or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">q3</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> or  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) &amp; (  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> or  not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) ) ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E33:46"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="gate_1.html#E34:46"><i><font color="Green" title="E73">A34</font></i></a>, <a class="txt" href="gate_1.html#E36:46"><i><font color="Green" title="E75">A36</font></i></a>, <a class="txt" href="gate_1.html#E37:46"><i><font color="Green" title="E76">A37</font></i></a>, <a class="txt" href="gate_1.html#E39:46"><i><font color="Green" title="E78">A39</font></i></a>, <a class="txt" href="gate_1.html#E40:46"><i><font color="Green" title="E79">A40</font></i></a>, <a class="txt" href="gate_1.html#E70:46"><i><font color="Green" title="E106">A67</font></i></a>, <a class="txt" href="gate_1.html#E73:46"><i><font color="Green" title="E107">A68</font></i></a></i>;<br/>
<a NAME="E76:46"/><b>then </b>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K38" title="GATE_1:func.38">CARR3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E44:46"><i><font color="Green" title="E83">A44</font></i></a>, <a class="txt" href="gate_1.html#E45:46"><i><font color="Green" title="E84">A45</font></i></a>, <a class="txt" href="gate_1.html#E55:46"><i><font color="Green" title="E94">A55</font></i></a>, <a class="txt" href="gate_1.html#E56:46"><i><font color="Green" title="E95">A56</font></i></a>, <a class="txt" href="gate_1.html#E57:46"><i><font color="Green" title="E96">A57</font></i></a>, <a class="txt" href="gate_1.html#E58:46"><i><font color="Green" title="E97">A58</font></i></a>, <a class="txt" href="gate_1.html#E59:46"><i><font color="Green" title="E98">A59</font></i></a>, <a class="ref" href="gate_1.html#T18" target="_self" title="GATE_1:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E77:46"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c41">s4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K39" title="GATE_1:func.39">ADD4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> )</span><br/>

<a NAME="E78:46"/>
( ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c20">q4</font> &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c17">q3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c14">q2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">q1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c21">p4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c18">p3</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c15">p2</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">p1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c23">cb1</font> ) ) iff ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">y4</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K38" title="GATE_1:func.38">CARR3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> ) or (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K38" title="GATE_1:func.38">CARR3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">x4</font> ) ) )
 
<b>by </b><i><a class="txt" href="gate_1.html#E13:46"><i><font color="Green" title="E52">A13</font></i></a>, <a class="txt" href="gate_1.html#E33:46"><i><font color="Green" title="E72">A33</font></i></a>, <a class="txt" href="gate_1.html#E34:46"><i><font color="Green" title="E73">A34</font></i></a>, <a class="txt" href="gate_1.html#E36:46"><i><font color="Green" title="E75">A36</font></i></a>, <a class="txt" href="gate_1.html#E37:46"><i><font color="Green" title="E76">A37</font></i></a>, <a class="txt" href="gate_1.html#E39:46"><i><font color="Green" title="E78">A39</font></i></a>, <a class="txt" href="gate_1.html#E40:46"><i><font color="Green" title="E79">A40</font></i></a>, <a class="txt" href="gate_1.html#E42:46"><i><font color="Green" title="E81">A42</font></i></a>, <a class="txt" href="gate_1.html#E43:46"><i><font color="Green" title="E82">A43</font></i></a>, <a class="txt" href="gate_1.html#E74:46"><i><font color="Green" title="E108">A69</font></i></a>, <a class="ref" href="gate_1.html#T4" target="_self" title="GATE_1:th.4">Th4</a></i>;<br/>
<a NAME="E79:46"/><b>then </b>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K38" title="GATE_1:func.38">CARR3</a> <font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_1.html#E60:46"><i><font color="Green" title="E99">A60</font></i></a>, <a class="txt" href="gate_1.html#E61:46"><i><font color="Green" title="E100">A61</font></i></a>, <a class="txt" href="gate_1.html#E62:46"><i><font color="Green" title="E101">A62</font></i></a>, <a class="txt" href="gate_1.html#E63:46"><i><font color="Green" title="E102">A63</font></i></a>, <a class="txt" href="gate_1.html#E64:46"><i><font color="Green" title="E103">A64</font></i></a>, <a class="ref" href="gate_1.html#T19" target="_self" title="GATE_1:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E80:46"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">c4</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K40" title="GATE_1:func.40">CARR4</a> <font color="Maroon" title="c8">x4</font>,<font color="Maroon" title="c9">y4</font>,<font color="Maroon" title="c6">x3</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">c1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
