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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">y0</font>, <font color="Maroon" title="c4">y1</font>, <font color="Maroon" title="c5">z0</font>, <font color="Maroon" title="c6">z1</font>, <font color="Maroon" title="c7">z2</font>, <font color="Maroon" title="c8">z3</font>, <font color="Maroon" title="c9">q00</font>, <font color="Maroon" title="c10">q01</font>, <font color="Maroon" title="c11">c01</font>, <font color="Maroon" title="c12">q11</font>, <font color="Maroon" title="c13">c11</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <!--HUMANRECFAILED-->( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">q00</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">q00</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">q01</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <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="c3">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="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">q01</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">c01</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <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="c3">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="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y0</font></span>)</span>,<span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">c01</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">q11</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <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">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c11">c01</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c11">c01</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">q11</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">c11</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <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">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c11">c01</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <span class="p1">(<span class="default"><a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c11">c01</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">c11</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">z0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">q00</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">q00</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">z0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">z1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">q01</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">q01</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">z1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">q11</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">q11</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">c11</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">c11</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> ) &amp; not <!--HUMANRECFAILED-->( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">z0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K1" title="GATE_5:func.1">MULT210</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K1" title="GATE_5:func.1">MULT210</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">z0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">z1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K2" title="GATE_5:func.2">MULT211</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K2" title="GATE_5:func.2">MULT211</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">z1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> ) ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green" title="E3">A1</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">q00</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> )
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green" title="E4">A2</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">q01</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <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="c3">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="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green" title="E5">A3</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c11">c01</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <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="c3">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="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>and </b><br/><a NAME="E4:4"/><i><font color="Green" title="E6">A4</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">q11</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K10" title="GATE_1:func.10">XOR3</a> <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">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c11">c01</font> )
 <b>and </b><br/><a NAME="E5:4"/><i><font color="Green" title="E7">A5</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">c11</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <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">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c11">c01</font> )
 <b>and </b><br/><a NAME="E6:4"/><i><font color="Green" title="E8">A6</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">z0</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c9">q00</font> )
 <b>and </b><br/><a NAME="E7:4"/><i><font color="Green" title="E9">A7</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">z1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c10">q01</font> )
 <b>and </b><br/><a NAME="E8:4"/><i><font color="Green" title="E10">A8</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">q11</font> )
 <b>and </b><br/><a NAME="E9:4"/><i><font color="Green" title="E11">A9</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">c11</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">z0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K1" title="GATE_5:func.1">MULT210</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K1" title="GATE_5:func.1">MULT210</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">z0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">z1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K2" title="GATE_5:func.2">MULT211</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K2" title="GATE_5:func.2">MULT211</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">z1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> ) )</span><br/>

<b>set </b><font color="Maroon" title="c14">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="c3">y0</font>;<br/>
<b>set </b><font color="Maroon" title="c15">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="c4">y1</font>;<br/>
<b>set </b><font color="Maroon" title="c16">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="c4">y1</font>;<br/>
<a NAME="E10:4"/><i><font color="Green" title="E12">A10</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c12">q11</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <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">y1</font></span>)</span>,<font color="Maroon" title="c11">c01</font> )
 
<b>by </b><i><a class="txt" href="gate_5.html#E4:4"><i><font color="Green" title="E6">A4</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="E11:4"/><i><font color="Green" title="E13">A11</font></i>: 
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c13">c11</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <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">y1</font></span>)</span>,<font color="Maroon" title="c11">c01</font> )
 
<b>by </b><i><a class="txt" href="gate_5.html#E5:4"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="gate_5.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E12:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c5">z0</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K1" title="GATE_5:func.1">MULT210</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> )
 <b>by </b><i><a class="txt" href="gate_5.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="gate_5.html#E6:4"><i><font color="Green" title="E8">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">z1</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K2" title="GATE_5:func.2">MULT211</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K2" title="GATE_5:func.2">MULT211</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">z1</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> ) )</span><br/>

<b>thus </b><a NAME="E13:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c6">z1</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K2" title="GATE_5:func.2">MULT211</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> )
 <b>by </b><i><a class="txt" href="gate_5.html#E2:4"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="gate_5.html#E7:4"><i><font color="Green" title="E9">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> ) &amp; (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> implies  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> ) )</span><br/>

<b>set </b><font color="Maroon" title="c17">m212</font> =  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font>;<br/>
<a NAME="E14:4"/>
 <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">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> <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="c2">x1</font>,<font color="Maroon" title="c4">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="c3">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="c4">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="E15:4"/><b>then </b>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K4" title="GATE_1:func.4">XOR2</a> <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">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="c3">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="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_5.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E16:4"/><b>then </b>
( ( (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font> &amp; not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <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="c3">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="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( not  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <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="c3">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="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> )
 
<b>by </b><i><a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E17:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c7">z2</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K3" title="GATE_5:func.3">MULT212</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> )
 <b>by </b><i><a class="txt" href="gate_5.html#E3:4"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="gate_5.html#E8:4"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="gate_5.html#E10:4"><i><font color="Green" title="E12">A10</font></i></a>, <a class="ref" href="gate_1.html#T8" title="GATE_1:th.8">GATE_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> )</span><br/>

<b>set </b><font color="Maroon" title="c18">m213</font> =  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font>;<br/>
<a NAME="E18:4"/>
 <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">x1</font>,<font color="Maroon" title="c4">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="c3">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="c4">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#D38" title="GATE_1:def.38">GATE_1:def 38</a></i>;<br/>
<a NAME="E19:4"/><b>then </b>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <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">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="c3">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="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="gate_5.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E20:4"/><b>then </b>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> iff (  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K2" title="GATE_1:func.2">AND2</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font> &amp;  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_1.html#K11" title="GATE_1:func.11">MAJ3</a> <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="c3">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="c4">y1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>;<br/>
<b>hence </b><a NAME="E21:4"/>
(  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a> <font color="Maroon" title="c8">z3</font> iff  <a href="gate_1.html#NR1" title="GATE_1:NR.1">$</a>  <a href="gate_5.html#K4" title="GATE_5:func.4">MULT213</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">y0</font> )
 <b>by </b><i><a class="txt" href="gate_5.html#E3:4"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="gate_5.html#E9:4"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="gate_5.html#E11:4"><i><font color="Green" title="E13">A11</font></i></a>, <a class="ref" href="gate_1.html#T6" title="GATE_1:th.6">GATE_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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