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<b>let </b><font color="Maroon" title="c1">x0</font>, <font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">y0</font>, <font color="Maroon" title="c5">y1</font>, <font color="Maroon" title="c6">y2</font>, <font color="Maroon" title="c7">z0</font>, <font color="Maroon" title="c8">z1</font>, <font color="Maroon" title="c9">z2</font>, <font color="Maroon" title="c10">z3</font>, <font color="Maroon" title="c11">z4</font>, <font color="Maroon" title="c12">z5</font>, <font color="Maroon" title="c13">q00</font>, <font color="Maroon" title="c14">q01</font>, <font color="Maroon" title="c15">q02</font>, <font color="Maroon" title="c16">c01</font>, <font color="Maroon" title="c17">c02</font>, <font color="Maroon" title="c18">q11</font>, <font color="Maroon" title="c19">q12</font>, <font color="Maroon" title="c20">c11</font>, <font color="Maroon" title="c21">c12</font>, <font color="Maroon" title="c22">q21</font>, <font color="Maroon" title="c23">q22</font>, <font color="Maroon" title="c24">c21</font>, <font color="Maroon" title="c25">c22</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-->( ( not <font color="Maroon" title="c13">q00</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c13">q00</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c14">q01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c14">q01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c15">q02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c15">q02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c18">q11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <font color="Maroon" title="c15">q02</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <font color="Maroon" title="c15">q02</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c18">q11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c15">q02</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c15">q02</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c19">q12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c19">q12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c22">q21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <font color="Maroon" title="c19">q12</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <font color="Maroon" title="c19">q12</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c22">q21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c24">c21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c19">q12</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c19">q12</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c24">c21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c23">q22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c24">c21</font>,<font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c24">c21</font>,<font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c23">q22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c25">c22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c24">c21</font>,<font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c24">c21</font>,<font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c25">c22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c7">z0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c13">q00</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c13">q00</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c7">z0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c8">z1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c14">q01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c14">q01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c8">z1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c18">q11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c18">q11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c22">q21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c22">q21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c23">q22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c23">q22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c25">c22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c25">c22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <!--HUMANRECFAILED-->( ( not <font color="Maroon" title="c7">z0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K5" title="GATE_5:func.5">MULT310</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K5" title="GATE_5:func.5">MULT310</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c7">z0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c8">z1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K6" title="GATE_5:func.6">MULT311</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K6" title="GATE_5:func.6">MULT311</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c8">z1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K10" title="GATE_5:func.10">MULT321</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K10" title="GATE_5:func.10">MULT321</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green" title="E2">A1</font></i>: 
( not <font color="Maroon" title="c13">q00</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green" title="E3">A2</font></i>: 
( not <font color="Maroon" title="c14">q01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E3:6"/><i><font color="Green" title="E4">A3</font></i>: 
( not <font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E4:6"/><i><font color="Green" title="E5">A4</font></i>: 
( not <font color="Maroon" title="c15">q02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E5:6"/><i><font color="Green" title="E6">A5</font></i>: 
( not <font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E6:6"/><i><font color="Green" title="E7">A6</font></i>: 
( not <font color="Maroon" title="c18">q11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <font color="Maroon" title="c15">q02</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E7:6"/><i><font color="Green" title="E8">A7</font></i>: 
( not <font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c15">q02</font>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E8:6"/><i><font color="Green" title="E9">A8</font></i>: 
( not <font color="Maroon" title="c19">q12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E9:6"/><i><font color="Green" title="E10">A9</font></i>: 
( not <font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E10:6"/><i><font color="Green" title="E11">A10</font></i>: 
( not <font color="Maroon" title="c22">q21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <font color="Maroon" title="c19">q12</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E11:6"/><i><font color="Green" title="E12">A11</font></i>: 
( not <font color="Maroon" title="c24">c21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <font color="Maroon" title="c19">q12</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E12:6"/><i><font color="Green" title="E13">A12</font></i>: 
( not <font color="Maroon" title="c23">q22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c24">c21</font>,<font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E13:6"/><i><font color="Green" title="E14">A13</font></i>: 
( not <font color="Maroon" title="c25">c22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c24">c21</font>,<font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E14:6"/><i><font color="Green" title="E15">A14</font></i>: 
( not <font color="Maroon" title="c7">z0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not <font color="Maroon" title="c13">q00</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E15:6"/><i><font color="Green" title="E16">A15</font></i>: 
( not <font color="Maroon" title="c8">z1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not <font color="Maroon" title="c14">q01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E16:6"/><i><font color="Green" title="E17">A16</font></i>: 
( not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not <font color="Maroon" title="c18">q11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E17:6"/><i><font color="Green" title="E18">A17</font></i>: 
( not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not <font color="Maroon" title="c22">q21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E18:6"/><i><font color="Green" title="E19">A18</font></i>: 
( not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not <font color="Maroon" title="c23">q22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E19:6"/><i><font color="Green" title="E20">A19</font></i>: 
( not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not <font color="Maroon" title="c25">c22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( not <font color="Maroon" title="c7">z0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K5" title="GATE_5:func.5">MULT310</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K5" title="GATE_5:func.5">MULT310</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c7">z0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c8">z1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K6" title="GATE_5:func.6">MULT311</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K6" title="GATE_5:func.6">MULT311</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c8">z1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K10" title="GATE_5:func.10">MULT321</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K10" title="GATE_5:func.10">MULT321</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )</span><br/>

<b>set </b><font color="Maroon" title="c26">x1y0</font> =  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font>;<br/>
<b>set </b><font color="Maroon" title="c27">x2y0</font> =  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font>;<br/>
<b>set </b><font color="Maroon" title="c28">x0y1</font> =  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font>;<br/>
<b>set </b><font color="Maroon" title="c29">x1y1</font> =  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>;<br/>
<b>set </b><font color="Maroon" title="c30">x2y1</font> =  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font>;<br/>
<b>set </b><font color="Maroon" title="c31">x0y2</font> =  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font>;<br/>
<b>set </b><font color="Maroon" title="c32">x1y2</font> =  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font>;<br/>
<b>set </b><font color="Maroon" title="c33">x2y2</font> =  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>;<br/>
<a NAME="E20:6"/><i><font color="Green" title="E21">A26</font></i>: 
( ( ( ( ( not <font color="Maroon" title="c15">q02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( <font color="Maroon" title="c15">q02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) &amp; <font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not <!--HUMANRECFAILED-->( not <font color="Maroon" title="c15">q02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c15">q02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) iff  not <font color="Maroon" title="c18">q11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="#E6:6"><i><font color="Green" title="E7">A6</font></i></a></i>;<br/>
<a NAME="E21:6"/><i><font color="Green" title="E22">A27</font></i>: 
( ( ( not <font color="Maroon" title="c15">q02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not <font color="Maroon" title="c16">c01</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c15">q02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) iff  not <font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="#E7:6"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/>
<a NAME="E22:6"/><i><font color="Green" title="E23">A28</font></i>: 
( ( ( ( ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or (  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) &amp; <font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) iff  not <font color="Maroon" title="c19">q12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="#E8:6"><i><font color="Green" title="E9">A8</font></i></a></i>;<br/>
<a NAME="E23:6"/><i><font color="Green" title="E24">A29</font></i>: 
( ( ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not <font color="Maroon" title="c17">c02</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) iff  not <font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="#E9:6"><i><font color="Green" title="E10">A9</font></i></a></i>;<br/>
<a NAME="E24:6"/><i><font color="Green" title="E25">A30</font></i>: 
( not <font color="Maroon" title="c22">q21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <font color="Maroon" title="c19">q12</font>,<font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="#E10:6"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="gate_5.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E25:6"/><i><font color="Green" title="E26">A31</font></i>: 
( not <font color="Maroon" title="c24">c21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( not <font color="Maroon" title="c19">q12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c20">c11</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 
<b>by </b><i><a class="txt" href="#E11:6"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<a NAME="E26:6"/><i><font color="Green" title="E27">A32</font></i>: 
( ( ( ( ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c24">c21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or (  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c24">c21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) &amp; <font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c24">c21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c24">c21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) iff  not <font color="Maroon" title="c23">q22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="#E12:6"><i><font color="Green" title="E13">A12</font></i></a></i>;<br/>
<a NAME="E27:6"/><i><font color="Green" title="E28">A33</font></i>: 
( ( ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c24">c21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not <font color="Maroon" title="c24">c21</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not <font color="Maroon" title="c21">c12</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) iff  not <font color="Maroon" title="c25">c22</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="#E13:6"><i><font color="Green" title="E14">A13</font></i></a></i>;<br/>
<b>thus </b><a NAME="E28:6"/>
( not <font color="Maroon" title="c7">z0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_5.html#K5" title="GATE_5:func.5">MULT310</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E1:6"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="#E14:6"><i><font color="Green" title="E15">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( not <font color="Maroon" title="c8">z1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K6" title="GATE_5:func.6">MULT311</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K6" title="GATE_5:func.6">MULT311</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c8">z1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K10" title="GATE_5:func.10">MULT321</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K10" title="GATE_5:func.10">MULT321</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )</span><br/>

<b>thus </b><a NAME="E29:6"/>
( not <font color="Maroon" title="c8">z1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_5.html#K6" title="GATE_5:func.6">MULT311</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E2:6"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E15:6"><i><font color="Green" title="E16">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K10" title="GATE_5:func.10">MULT321</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K10" title="GATE_5:func.10">MULT321</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )</span><br/>

<b>set </b><font color="Maroon" title="c34">m312</font> =  <a href="gate_5.html#K7" title="GATE_5:func.7">MULT312</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font>;<br/>
<a NAME="E30:6"/><i><font color="Green" title="E29">B</font></i>: 
 <a href="gate_5.html#K7" title="GATE_5:func.7">MULT312</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gate_1.html#D37" title="GATE_1:def.37">GATE_1:def 37</a></i>;<br/>
<a NAME="E31:6"/><b>then </b><i><font color="Green" title="E30">A34</font></i>: 
( ( ( ( ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or (  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) &amp;  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) iff  not  <a href="gate_5.html#K7" title="GATE_5:func.7">MULT312</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
;<br/>
<a NAME="E32:6"/><i><font color="Green" title="E31">A35</font></i>: 
( ( ( ( ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or (  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) &amp; (  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) or ( not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <!--HUMANRECFAILED-->(  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) iff  not  <a href="gate_5.html#K7" title="GATE_5:func.7">MULT312</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="#E30:6"><i><font color="Green" title="E29">B</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c35">m321</font> =  <a href="gate_5.html#K10" title="GATE_5:func.10">MULT321</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font>;<br/>
<b>thus </b><a NAME="E33:6"/>
( not <font color="Maroon" title="c9">z2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_5.html#K10" title="GATE_5:func.10">MULT321</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E3:6"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="#E16:6"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="#E4:6"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="#E20:6"><i><font color="Green" title="E21">A26</font></i></a>, <a class="txt" href="#E31:6"><i><font color="Green" title="E30">A34</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )</span><br/>

<b>set </b><font color="Maroon" title="c36">m313</font> =  <a href="gate_5.html#K8" title="GATE_5:func.8">MULT313</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font>;<br/>
 <a href="gate_5.html#K8" title="GATE_5:func.8">MULT313</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> = 
 <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K36" title="GATE_1:func.36">CARR2</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="gate_1.html#D39" title="GATE_1:def.39">GATE_1:def 39</a></i>
<br/>.= 
 <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p3">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p3">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gate_1.html#D38" title="GATE_1:def.38">GATE_1:def 38</a></i>
;<br/>
<a NAME="E35:6"/><b>then </b><i><font color="Green" title="E32">A38</font></i>: 
( ( ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; not <!--HUMANRECFAILED-->( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) or (  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; ( ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) ) ) iff  not  <a href="gate_5.html#K8" title="GATE_5:func.8">MULT313</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
;<br/>
<b>set </b><font color="Maroon" title="c37">m322</font> =  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font>;<br/>
<a NAME="E36:6"/>
 <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_5.html#K8" title="GATE_5:func.8">MULT313</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p2">(<span class="default"><a href="gate_5.html#K7" title="GATE_5:func.7">MULT312</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="gate_1.html#D37" title="GATE_1:def.37">GATE_1:def 37</a></i>;<br/>
<b>hence </b><a NAME="E37:6"/>
( not <font color="Maroon" title="c10">z3</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_5.html#K11" title="GATE_5:func.11">MULT322</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E17:6"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="#E3:6"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="#E4:6"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="#E5:6"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="#E21:6"><i><font color="Green" title="E22">A27</font></i></a>, <a class="txt" href="#E22:6"><i><font color="Green" title="E23">A28</font></i></a>, <a class="txt" href="#E24:6"><i><font color="Green" title="E25">A30</font></i></a>, <a class="txt" href="#E32:6"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="#E35:6"><i><font color="Green" title="E32">A38</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) &amp; ( not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )</span><br/>

<b>set </b><font color="Maroon" title="c38">m314</font> =  <a href="gate_5.html#K9" title="GATE_5:func.9">MULT314</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font>;<br/>
 <a href="gate_5.html#K9" title="GATE_5:func.9">MULT314</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> = 
 <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K36" title="GATE_1:func.36">CARR2</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="gate_1.html#D40" title="GATE_1:def.40">GATE_1:def 40</a></i>
<br/>.= 
 <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p3">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p3">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gate_1.html#D38" title="GATE_1:def.38">GATE_1:def 38</a></i>
;<br/>
<a NAME="E39:6"/><b>then </b><i><font color="Green" title="E33">A39</font></i>: 
( not  <a href="gate_5.html#K9" title="GATE_5:func.9">MULT314</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; ( ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) or ( not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c5">y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) ) )
 
;<br/>
<b>set </b><font color="Maroon" title="c39">m323</font> =  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font>;<br/>
 <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> = 
 <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_5.html#K9" title="GATE_5:func.9">MULT314</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K36" title="GATE_1:func.36">CARR2</a> <span class="p2">(<span class="default"><a href="gate_5.html#K8" title="GATE_5:func.8">MULT313</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_5.html#K7" title="GATE_5:func.7">MULT312</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="gate_1.html#D39" title="GATE_1:def.39">GATE_1:def 39</a></i>
<br/>.= 
 <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_5.html#K9" title="GATE_5:func.9">MULT314</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p2">(<span class="default"><a href="gate_5.html#K8" title="GATE_5:func.8">MULT313</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p3">(<span class="default"><a href="gate_5.html#K7" title="GATE_5:func.7">MULT312</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p3">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gate_1.html#D38" title="GATE_1:def.38">GATE_1:def 38</a></i>
;<br/>
<b>hence </b><a NAME="E41:6"/>
( not <font color="Maroon" title="c11">z4</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_5.html#K12" title="GATE_5:func.12">MULT323</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E18:6"><i><font color="Green" title="E19">A18</font></i></a>, <a class="txt" href="#E3:6"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="#E4:6"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="#E5:6"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="#E21:6"><i><font color="Green" title="E22">A27</font></i></a>, <a class="txt" href="#E22:6"><i><font color="Green" title="E23">A28</font></i></a>, <a class="txt" href="#E23:6"><i><font color="Green" title="E24">A29</font></i></a>, <a class="txt" href="#E25:6"><i><font color="Green" title="E26">A31</font></i></a>, <a class="txt" href="#E26:6"><i><font color="Green" title="E27">A32</font></i></a>, <a class="txt" href="#E32:6"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="#E35:6"><i><font color="Green" title="E32">A38</font></i></a>, <a class="txt" href="#E39:6"><i><font color="Green" title="E33">A39</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/>

<b>set </b><font color="Maroon" title="c40">m324</font> =  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font>;<br/>
 <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> = 
 <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_5.html#K9" title="GATE_5:func.9">MULT314</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K36" title="GATE_1:func.36">CARR2</a> <span class="p2">(<span class="default"><a href="gate_5.html#K8" title="GATE_5:func.8">MULT313</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_5.html#K7" title="GATE_5:func.7">MULT312</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="gate_1.html#D40" title="GATE_1:def.40">GATE_1:def 40</a></i>
<br/>.= 
 <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_5.html#K9" title="GATE_5:func.9">MULT314</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p2">(<span class="default"><a href="gate_5.html#K8" title="GATE_5:func.8">MULT313</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p2">(<span class="default"><a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p3">(<span class="default"><a href="gate_5.html#K7" title="GATE_5:func.7">MULT312</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font></span>)</span>,<span class="p3">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gate_1.html#D38" title="GATE_1:def.38">GATE_1:def 38</a></i>
;<br/>
<b>hence </b><a NAME="E43:6"/>
( not <font color="Maroon" title="c12">z5</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> iff  not  <a href="gate_5.html#K13" title="GATE_5:func.13">MULT324</a> <font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="#E19:6"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="#E3:6"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="#E4:6"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="#E5:6"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="#E21:6"><i><font color="Green" title="E22">A27</font></i></a>, <a class="txt" href="#E22:6"><i><font color="Green" title="E23">A28</font></i></a>, <a class="txt" href="#E23:6"><i><font color="Green" title="E24">A29</font></i></a>, <a class="txt" href="#E25:6"><i><font color="Green" title="E26">A31</font></i></a>, <a class="txt" href="#E27:6"><i><font color="Green" title="E28">A33</font></i></a>, <a class="txt" href="#E32:6"><i><font color="Green" title="E31">A35</font></i></a>, <a class="txt" href="#E35:6"><i><font color="Green" title="E32">A38</font></i></a>, <a class="txt" href="#E39:6"><i><font color="Green" title="E33">A39</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
